Number Series, Coding and DecodingMCQMTP Sep 24 Series IIQuestion 2080 of 217
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Find next term of the series $1, 3, 4, 8, 15, 27, ?$

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

✅ Option c — $50$

All Options:

  • A$37$
  • B$44$
  • C$50$
  • D$55$

Detailed Solution & Explanation

To find the next term of the series $1, 3, 4, 8, 15, 27, ?,letusanalyzetherelationofeachtermtoitsprecedingterms:<br><br>Weobservethatstartingfromthefourthterm(\displaystyle , let us analyze the relation of each term to its preceding terms:<br><br>We observe that starting from the fourth term (8),eachtermisthesumofthethreeprecedingterms:<br>−4thterm:\displaystyle ), each term is the sum of the three preceding terms:<br>- 4th term:1 + 3 + 4 = 8<br>−5thterm:\displaystyle <br>- 5th term:3 + 4 + 8 = 15<br>−6thterm:\displaystyle <br>- 6th term:4 + 8 + 15 = 27<br><br>ThisisthecharacteristicruleofaTribonacci−likesequence:<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><msub><mi>T</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>T</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>T</mi><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></mrow><annotationencoding="application/x−tex">Tn=Tn−1+Tn−2+Tn−3</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:0.8917em;vertical−align:−0.2083em;"></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:0.3011em;"><spanstyle="top:−2.55em;margin−left:−0.1389em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span><spanclass="mbinmtight">−</span><spanclass="mordmtight">1</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2083em;"><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:0.8917em;vertical−align:−0.2083em;"></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:0.3011em;"><spanstyle="top:−2.55em;margin−left:−0.1389em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span><spanclass="mbinmtight">−</span><spanclass="mordmtight">2</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2083em;"><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:0.8917em;vertical−align:−0.2083em;"></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:0.3011em;"><spanstyle="top:−2.55em;margin−left:−0.1389em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span><spanclass="mbinmtight">−</span><spanclass="mordmtight">3</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span><br>Followingthislogicalprogression,thenextterm(7thterm)mustbethesumofthe4th,5th,and6thterms:<br>−7thterm:\displaystyle <br><br>This is the characteristic rule of a Tribonacci-like sequence:<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><msub><mi>T</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>T</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>T</mi><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">T_n = T_{n-1} + T_{n-2} + T_{n-3}</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:0.8917em;vertical-align:-0.2083em;"></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:0.3011em;"><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 mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><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:0.8917em;vertical-align:-0.2083em;"></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:0.3011em;"><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 mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><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:0.8917em;vertical-align:-0.2083em;"></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:0.3011em;"><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 mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span><br>Following this logical progression, the next term (7th term) must be the sum of the 4th, 5th, and 6th terms:<br>- 7th term:8 + 15 + 27 = 50\displaystyle <br><br>Therefore, the mathematically correct answer is **&#36;50**, which corresponds to **Option C**.

*Note on Answer Key Discrepancy:* The textbook answer key lists **Option D** ($55$) as the correct answer. However, $55$ does not satisfy the clear, established Tribonacci progression of the sequence, whereas $50$ completes the mathematical logic perfectly. This is a clear typographical error in the official answer key, and the correct option is **Option C**.

Hence, **Option C** 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.

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Exam Strategy Tip

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

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