Number Series, Coding and DecodingMCQMTP Dec 2023 Series IQuestion 2073 of 217
All Questions

Find next number of the series 7,23,47,119,167\displaystyle 7, 23, 47, 119, 167

Options

A211\displaystyle 211
B223\displaystyle 223
C287\displaystyle 287
D319\displaystyle 319
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Correct Answer

✅ Option c — 287\displaystyle 287

All Options:

  • A211\displaystyle 211
  • B223\displaystyle 223
  • C287\displaystyle 287
  • D319\displaystyle 319

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Detailed Solution & Explanation

To find the next number in the series 7,23,47,119,167,?\displaystyle 7, 23, 47, 119, 167, ?, let us analyze each term's relationship to perfect squares:

Let Tn\displaystyle T_n represent the n\displaystyle n-th term. We can express each term using the formula:
Tn=(pn)2−2T_n = (p_n)^2 - 2
where pn\displaystyle p_n represents the sequence of consecutive prime numbers starting from 3\displaystyle 3 (3,5,7,11,13,…\displaystyle 3, 5, 7, 11, 13, \dots). Note that 9\displaystyle 9 is omitted because it is a composite number, not a prime number.

Let us compute the terms to verify this pattern:
- 1st term (p1=3\displaystyle p_1 = 3): 32−2=9−2=7\displaystyle 3^2 - 2 = 9 - 2 = 7
- 2nd term (p2=5\displaystyle p_2 = 5): 52−2=25−2=23\displaystyle 5^2 - 2 = 25 - 2 = 23
- 3rd term (p3=7\displaystyle p_3 = 7): 72−2=49−2=47\displaystyle 7^2 - 2 = 49 - 2 = 47
- 4th term (p4=11\displaystyle p_4 = 11): 112−2=121−2=119\displaystyle 11^2 - 2 = 121 - 2 = 119
- 5th term (p5=13\displaystyle p_5 = 13): 132−2=169−2=167\displaystyle 13^2 - 2 = 169 - 2 = 167

Following this logical progression, the next prime number after 13\displaystyle 13 is 17\displaystyle 17. Therefore, the 6th term must be:
- 6th term (p6=17\displaystyle p_6 = 17): 172−2=289−2=287\displaystyle 17^2 - 2 = 289 - 2 = 287

This mathematical proof uniquely demonstrates that the next term in the series is **287\displaystyle 287**, which corresponds to **Option C**.

*Note on Answer Key Discrepancy:* The textbook/exam answer key lists **Option A** (211\displaystyle 211) as the correct answer. However, 211\displaystyle 211 does not correspond to any logical progression or prime square pattern, whereas 287\displaystyle 287 fits the consecutive prime-squared minus two formula 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.

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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