Sequence and SeriesMCQMTP Nov 19Question 1790 of 213
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The first term of an A.P. is $100$ and the sum of whose first $6termsis\displaystyle terms is5$ times the sum of the next $6$ terms, then the c.d. is:

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Correct Answer

✅ Option d — none of these

All Options:

  • A10
  • B10
  • C5
  • Dnone of these

Detailed Solution & Explanation

Given the first term a=100\displaystyle a = 100. Let d\displaystyle d be the common difference.
The sum of the first 6\displaystyle 6 terms (S6\displaystyle S_6) is:
S6=62[2a+5d]=3[2(100)+5d]=600+15dS_6 = 62\displaystyle \frac{6}{2} [2a + 5d] = 3 [2(100) + 5d] = 600 + 15d

The sum of the first $12terms(\displaystyle terms (S_{12})is:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msub><mi>S</mi><mn>12</mn></msub><mo>=</mo><mfrac><mn>12</mn><mn>2</mn></mfrac><mostretchy="false">[</mo><mn>2</mn><mi>a</mi><mo>+</mo><mn>11</mn><mi>d</mi><mostretchy="false">]</mo><mo>=</mo><mn>6</mn><mostretchy="false">[</mo><mn>2</mn><mostretchy="false">(</mo><mn>100</mn><mostretchy="false">)</mo><mo>+</mo><mn>11</mn><mi>d</mi><mostretchy="false">]</mo><mo>=</mo><mn>1200</mn><mo>+</mo><mn>66</mn><mi>d</mi></mrow><annotationencoding="application/x−tex">S12=122[2a+11d]=6[2(100)+11d]=1200+66d</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8333em;vertical−align:−0.15em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:−0.0576em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">12</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0074em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">12</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mopen">[</span><spanclass="mord">2</span><spanclass="mordmathnormal">a</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">11</span><spanclass="mordmathnormal">d</span><spanclass="mclose">]</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">6</span><spanclass="mopen">[</span><spanclass="mord">2</span><spanclass="mopen">(</span><spanclass="mord">100</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">11</span><spanclass="mordmathnormal">d</span><spanclass="mclose">]</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">1200</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mord">66</span><spanclass="mordmathnormal">d</span></span></span></span></span><br><br>Thesumofthenext\displaystyle ) is:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>S</mi><mn>12</mn></msub><mo>=</mo><mfrac><mn>12</mn><mn>2</mn></mfrac><mo stretchy="false">[</mo><mn>2</mn><mi>a</mi><mo>+</mo><mn>11</mn><mi>d</mi><mo stretchy="false">]</mo><mo>=</mo><mn>6</mn><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">(</mo><mn>100</mn><mo stretchy="false">)</mo><mo>+</mo><mn>11</mn><mi>d</mi><mo stretchy="false">]</mo><mo>=</mo><mn>1200</mn><mo>+</mo><mn>66</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">S_{12} = 122\displaystyle \frac{12}{2} [2a + 11d] = 6 [2(100) + 11d] = 1200 + 66d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">[</span><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">11</span><span class="mord mathnormal">d</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">6</span><span class="mopen">[</span><span class="mord">2</span><span class="mopen">(</span><span class="mord">100</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">11</span><span class="mord mathnormal">d</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1200</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">66</span><span class="mord mathnormal">d</span></span></span></span></span><br><br>The sum of the next6termsis:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msub><mi>S</mi><mrow><mtext>next </mtext><mn>6</mn></mrow></msub><mo>=</mo><msub><mi>S</mi><mn>12</mn></msub><mo>−</mo><msub><mi>S</mi><mn>6</mn></msub><mo>=</mo><mostretchy="false">(</mo><mn>1200</mn><mo>+</mo><mn>66</mn><mi>d</mi><mostretchy="false">)</mo><mo>−</mo><mostretchy="false">(</mo><mn>600</mn><mo>+</mo><mn>15</mn><mi>d</mi><mostretchy="false">)</mo><mo>=</mo><mn>600</mn><mo>+</mo><mn>51</mn><mi>d</mi></mrow><annotationencoding="application/x−tex">Snext 6=S12−S6=(1200+66d)−(600+15d)=600+51d</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8333em;vertical−align:−0.15em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:−0.0576em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordtextmtight"><spanclass="mordmtight">next </span></span><spanclass="mordmtight">6</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8333em;vertical−align:−0.15em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:−0.0576em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">12</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8333em;vertical−align:−0.15em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:−0.0576em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">6</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">1200</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">66</span><spanclass="mordmathnormal">d</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">600</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">15</span><spanclass="mordmathnormal">d</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">600</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mord">51</span><spanclass="mordmathnormal">d</span></span></span></span></span><br><br>Wearegiventhat\displaystyle terms is:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>S</mi><mrow><mtext>next </mtext><mn>6</mn></mrow></msub><mo>=</mo><msub><mi>S</mi><mn>12</mn></msub><mo>−</mo><msub><mi>S</mi><mn>6</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mn>1200</mn><mo>+</mo><mn>66</mn><mi>d</mi><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mn>600</mn><mo>+</mo><mn>15</mn><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mn>600</mn><mo>+</mo><mn>51</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">S_{next \displaystyle \text{next } 6} = S_{12} - S_6 = (1200 + 66d) - (600 + 15d) = 600 + 51d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">next </span></span><span class="mord mtight">6</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1200</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">66</span><span class="mord mathnormal">d</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">600</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">15</span><span class="mord mathnormal">d</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">600</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">51</span><span class="mord mathnormal">d</span></span></span></span></span><br><br>We are given thatS_6 = 5 ⋅\displaystyle \cdot S_{next \displaystyle \text{next } 6}:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>600</mn><mo>+</mo><mn>15</mn><mi>d</mi><mo>=</mo><mn>5</mn><mostretchy="false">(</mo><mn>600</mn><mo>+</mo><mn>51</mn><mi>d</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">600+15d=5(600+51d)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">600</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mord">15</span><spanclass="mordmathnormal">d</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">5</span><spanclass="mopen">(</span><spanclass="mord">600</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">51</span><spanclass="mordmathnormal">d</span><spanclass="mclose">)</span></span></span></span></span><br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>600</mn><mo>+</mo><mn>15</mn><mi>d</mi><mo>=</mo><mn>3000</mn><mo>+</mo><mn>255</mn><mi>d</mi></mrow><annotationencoding="application/x−tex">600+15d=3000+255d</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">600</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mord">15</span><spanclass="mordmathnormal">d</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">3000</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mord">255</span><spanclass="mordmathnormal">d</span></span></span></span></span><br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mo>−</mo><mn>2400</mn><mo>=</mo><mn>240</mn><mi>d</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>d</mi><mo>=</mo><mo>−</mo><mn>10</mn></mrow><annotationencoding="application/x−tex">−2400=240d  ⟹  d=−10</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">2400</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7184em;vertical−align:−0.024em;"></span><spanclass="mord">240</span><spanclass="mordmathnormal">d</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">⟹</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">d</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">10</span></span></span></span></span><br>So,thecommondifferenceis\displaystyle :<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>600</mn><mo>+</mo><mn>15</mn><mi>d</mi><mo>=</mo><mn>5</mn><mo stretchy="false">(</mo><mn>600</mn><mo>+</mo><mn>51</mn><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">600 + 15d = 5(600 + 51d)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">600</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">15</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mopen">(</span><span class="mord">600</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">51</span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span></span><br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>600</mn><mo>+</mo><mn>15</mn><mi>d</mi><mo>=</mo><mn>3000</mn><mo>+</mo><mn>255</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">600 + 15d = 3000 + 255d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">600</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">15</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">255</span><span class="mord mathnormal">d</span></span></span></span></span><br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>−</mo><mn>2400</mn><mo>=</mo><mn>240</mn><mi>d</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>d</mi><mo>=</mo><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">-2400 = 240d \implies d = -10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">2400</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7184em;vertical-align:-0.024em;"></span><span class="mord">240</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">10</span></span></span></span></span><br>So, the common difference is-10.<br>Hence,∗∗OptionD∗∗(sinceitis\displaystyle .<br>Hence, **Option D** (since it is-10$, which is 'none of these' if options only list positive 10) or **Option A** (magnitude 10) is the correct answer.

About This Chapter: Sequence and Series

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Arithmetic & Geometric Progressions

This chapter covers Arithmetic Progressions (AP) and Geometric Progressions (GP). Students learn how to find the nth term, sum of n terms, arithmetic/geometric means, and sum to infinity of a GP.

View Official ICAI Syllabus

Exam Strategy Tip

For complex 'sum of series' questions, a great hack is to substitute n = 1 and n = 2 into the question and the options to see which one matches, completely bypassing the formula.

Related Comparison Tables

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More Questions from Sequence and Series

Sum of progression (a+b)\displaystyle (a+b), (a−b)\displaystyle (a-b), (a−3b)...n\displaystyle (a-3b)...n term is

Find the sum of first twenty-five terms of A.P. series whose nth\displaystyle n^{th} term is (n+25)\displaystyle \left(n+25\displaystyle \frac{2}{5}\right)

The first and fifth term of an A.P. of $40termsare\displaystyle terms are-29and\displaystyle and-15$ respectively. Find the sum of all positive terms of this A.P

If the common difference of an AP equals to the first term, then the ratio of its mth\displaystyle m^{th} term and nth\displaystyle n^{th} term is:

Find the value of 1+2+3+...+105\displaystyle 1+2+3+...+105

The first and last terms of an arithmetic progression are 5\displaystyle 5 and $905$. Sum of the terms is $45,955$. The number of terms is

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