Number Series, Coding, Odd Man OutMTP Dec 22 Series IIQuestion 2066 of 217
All Questions

Find the missing term of the following series 3,7,15,?,63,127\displaystyle 3, 7, 15, ?, 63, 127

Options

A30\displaystyle 30
B31\displaystyle 31
C47\displaystyle 47
D52\displaystyle 52
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Correct Answer

✅ Option b — 31\displaystyle 31

All Options:

  • A30\displaystyle 30
  • B31\displaystyle 31
  • C47\displaystyle 47
  • D52\displaystyle 52

Detailed Solution & Explanation

To find the missing term of the series 3,7,15,?,63,127\displaystyle 3, 7, 15, ?, 63, 127, we can use two different mathematical methods:

**Method 1: Multiplying by 2 and adding 1**
Let us analyze the relationship between consecutive terms:
- 1st to 2nd term: 3×2+1=7\displaystyle 3 \times 2 + 1 = 7
- 2nd to 3rd term: 7×2+1=15\displaystyle 7 \times 2 + 1 = 15
- 3rd to 4th term (missing term): 15×2+1=31\displaystyle 15 \times 2 + 1 = 31
- 4th to 5th term: 31×2+1=63\displaystyle 31 \times 2 + 1 = 63
- 5th to 6th term: 63×2+1=127\displaystyle 63 \times 2 + 1 = 127

**Method 2: Powers of 2 minus 1**
We can express each term as Tn=2n+1−1\displaystyle T_n = 2^{n+1} - 1:
- T1=22−1=3\displaystyle T_1 = 2^2 - 1 = 3
- T2=23−1=7\displaystyle T_2 = 2^3 - 1 = 7
- T3=24−1=15\displaystyle T_3 = 2^4 - 1 = 15
- T4=25−1=31\displaystyle T_4 = 2^5 - 1 = 31 (missing term)
- T5=26−1=63\displaystyle T_5 = 2^6 - 1 = 63
- T6=27−1=127\displaystyle T_6 = 2^7 - 1 = 127

Both methods consistently and mathematically prove that the missing term is **31\displaystyle 31**, which corresponds to **Option B**.

*Note on Answer Key Discrepancy:* The textbook/exam answer key lists **Option A** (30\displaystyle 30) as the correct answer. However, 30\displaystyle 30 violates both mathematical progressions of the sequence, whereas 31\displaystyle 31 is logically and mathematically perfect. This is a clear typographical error in the official answer key, and the correct option is **Option B**.

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

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