Sequence and SeriesMCQMTP June 22Question 1801 of 213
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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

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — 1705

All Options:

  • A1605
  • B1705
  • C1805
  • DNone of these

Detailed Solution & Explanation

Let the first term be a=−29\displaystyle a = -29 and the common difference be d\displaystyle d.
We are given:
t5=a+4d=−15t_5 = a + 4d = -15
−29+4d=−15  ⟹  4d=14  ⟹  d=3.5-29 + 4d = -15 \implies 4d = 14 \implies d = 3.5

Let us find which terms are positive:
tk=a+(k−1)d>0t_k = a + (k-1)d > 0
−29+(k−1)3.5>0-29 + (k-1)3.5 > 0
(k−1)3.5>29  ⟹  k−1>8.28  ⟹  k>9.28(k-1)3.5 > 29 \implies k-1 > 8.28 \implies k > 9.28
So the first positive term is the $10^{th\displaystyle \text{th}}term(\displaystyle term (t_{10}\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><msub><mi>t</mi><mn>10</mn></msub><mo>=</mo><mo>−</mo><mn>29</mn><mo>+</mo><mn>9</mn><mo stretchy="false">(</mo><mn>3.5</mn><mo stretchy="false">)</mo><mo>=</mo><mn>2.5</mn></mrow><annotation encoding="application/x-tex">t_{10} = -29 + 9(3.5) = 2.5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</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:0em;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">10</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:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">29</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">9</span><span class="mopen">(</span><span class="mord">3.5</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:0.6444em;"></span><span class="mord">2.5</span></span></span></span></span><br><br>The number of positive terms is &#36;40 - 9 = 31 terms.
The last term is t40\displaystyle t_{40}:
t40=a+39d=−29+39(3.5)=−29+136.5=107.5t_{40} = a + 39d = -29 + 39(3.5) = -29 + 136.5 = 107.5

Sum of these $31$ positive terms is:
Spos=312[t10+t40]S_{pos\displaystyle \text{pos}} = 312\displaystyle \frac{31}{2} [t_{10} + t_{40}]
Spos=312[2.5+107.5]=312[110]=31⋅55=1705S_{pos\displaystyle \text{pos}} = 312\displaystyle \frac{31}{2} [2.5 + 107.5] = 312\displaystyle \frac{31}{2} [110] = 31 ⋅\displaystyle \cdot 55 = 1705
Hence, **Option B** 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.

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