Number Series, Coding and DecodingMCQMTP Dec 22 Series IIQuestion 2064 of 217
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Find the missing term of the following series $3, 15, ?, 63, 99, 143$

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

✅ Option b — $35$

All Options:

  • A$27$
  • B$35$
  • C$45$
  • D$56$

Detailed Solution & Explanation

To find the missing term of the series $3, 15, ?, 63, 99, 143,letusanalyzetherelationofeachtermtoperfectsquaresofconsecutiveevennumbers:<br><br>Let\displaystyle , let us analyze the relation of each term to perfect squares of consecutive even numbers:<br><br>LetT_nrepresentthe\displaystyle represent then−thtermoftheseries.Thelogicalformulaforthetermsis:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msub><mi>T</mi><mi>n</mi></msub><mo>=</mo><mostretchy="false">(</mo><mn>2</mn><mi>n</mi><msup><mostretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">Tn=(2n)2−1</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:0.13889em;">T</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1514em;"><spanstyle="top:−2.55em;margin−left:−0.1389em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">n</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:1.1141em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">2</span><spanclass="mordmathnormal">n</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></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:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span><br>Letuscomputeeachtermtoverifythispattern:<br>−1stterm:\displaystyle -th term of the series. The logical formula for the 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>T</mi><mi>n</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">T_n = (2n)^2 - 1</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:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</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:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></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:0.6444em;"></span><span class="mord">1</span></span></span></span></span><br>Let us compute each term to verify this pattern:<br>- 1st term:T_1 = (2 ×\displaystyle \times 1)^2 - 1 = 4 - 1 = 3<br>−2ndterm:\displaystyle <br>- 2nd term:T_2 = (2 ×\displaystyle \times 2)^2 - 1 = 16 - 1 = 15<br>−3rdterm(missingterm):\displaystyle <br>- 3rd term (missing term):T_3 = (2 ×\displaystyle \times 3)^2 - 1 = 36 - 1 = 35<br>−4thterm:\displaystyle <br>- 4th term:T_4 = (2 ×\displaystyle \times 4)^2 - 1 = 64 - 1 = 63<br>−5thterm:\displaystyle <br>- 5th term:T_5 = (2 ×\displaystyle \times 5)^2 - 1 = 100 - 1 = 99<br>−6thterm:\displaystyle <br>- 6th term:T_6 = (2 ×\displaystyle \times 6)^2 - 1 = 144 - 1 = 143<br><br>∗∗AlternativeMethod(Differences):∗∗<br>Wecanalsolookatthedifferencesbetweenconsecutiveterms:<br>−\displaystyle <br><br>**Alternative Method (Differences):**<br>We can also look at the differences between consecutive terms:<br>-T_2 - T_1 = 15 - 3 = 12<br>−\displaystyle <br>-T_3 - T_2 = 35 - 15 = 20(differenceof\displaystyle (difference of+8from12)<br>−\displaystyle from 12)<br>-T_4 - T_3 = 63 - 35 = 28(differenceof\displaystyle (difference of+8from20)<br>−\displaystyle from 20)<br>-T_5 - T_4 = 99 - 63 = 36(differenceof\displaystyle (difference of+8from28)<br>−\displaystyle from 28)<br>-T_6 - T_5 = 143 - 99 = 44(differenceof\displaystyle (difference of+8\displaystyle from 36)<br>The difference sequence &#36;12, 20, 28, 36, 44 increases by exactly 8\displaystyle 8 each time, which mathematically proves that the missing term is $35\displaystyle .<br><br>*Note on Answer Key Discrepancy:* The textbook answer key lists **Option D** (&#36;56) as correct. However, $56$ does not follow any logical, consistent mathematical progression with the rest of the terms. The value $35$ fits perfectly under both the squared formula and the difference progression, making Option B the correct logical and mathematical answer. The selection of Option D in the answer key is a typographical error.

Hence, **Option B** is the correct answer.

About This Chapter: Number Series, Coding and Decoding

Paper

Paper 3: Quantitative Aptitude

Weightage

5 Marks

Key Topics

Number series, Coding & Decoding, Odd man out

This chapter covers Number series, Coding & Decoding, Odd man out and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5 Marks weightage. Focus on understanding core concepts rather than memorizing.

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