Number Series, Coding, Odd Man OutMTP Dec 22 Series IIQuestion 2064 of 217
All Questions

Find the missing term of the following series 3,15,?,63,99,143\displaystyle 3, 15, ?, 63, 99, 143

Options

A27\displaystyle 27
B35\displaystyle 35
C45\displaystyle 45
D56\displaystyle 56
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Correct Answer

✅ Option b — 35\displaystyle 35

All Options:

  • A27\displaystyle 27
  • B35\displaystyle 35
  • C45\displaystyle 45
  • D56\displaystyle 56

Detailed Solution & Explanation

To find the missing term of the series 3,15,?,63,99,143\displaystyle 3, 15, ?, 63, 99, 143, let us analyze the relation of each term to perfect squares of consecutive even numbers:

Let Tn\displaystyle T_n represent the n\displaystyle n-th term of the series. The logical formula for the terms is:
Tn=(2n)2−1T_n = (2n)^2 - 1
Let us compute each term to verify this pattern:
- 1st term: T1=(2×1)2−1=4−1=3\displaystyle T_1 = (2 \times 1)^2 - 1 = 4 - 1 = 3
- 2nd term: T2=(2×2)2−1=16−1=15\displaystyle T_2 = (2 \times 2)^2 - 1 = 16 - 1 = 15
- 3rd term (missing term): T3=(2×3)2−1=36−1=35\displaystyle T_3 = (2 \times 3)^2 - 1 = 36 - 1 = 35
- 4th term: T4=(2×4)2−1=64−1=63\displaystyle T_4 = (2 \times 4)^2 - 1 = 64 - 1 = 63
- 5th term: T5=(2×5)2−1=100−1=99\displaystyle T_5 = (2 \times 5)^2 - 1 = 100 - 1 = 99
- 6th term: T6=(2×6)2−1=144−1=143\displaystyle T_6 = (2 \times 6)^2 - 1 = 144 - 1 = 143

**Alternative Method (Differences):**
We can also look at the differences between consecutive terms:
- T2−T1=15−3=12\displaystyle T_2 - T_1 = 15 - 3 = 12
- T3−T2=35−15=20\displaystyle T_3 - T_2 = 35 - 15 = 20 (difference of +8\displaystyle +8 from 12)
- T4−T3=63−35=28\displaystyle T_4 - T_3 = 63 - 35 = 28 (difference of +8\displaystyle +8 from 20)
- T5−T4=99−63=36\displaystyle T_5 - T_4 = 99 - 63 = 36 (difference of +8\displaystyle +8 from 28)
- T6−T5=143−99=44\displaystyle T_6 - T_5 = 143 - 99 = 44 (difference of +8\displaystyle +8 from 36)
The difference sequence 12,20,28,36,44\displaystyle 12, 20, 28, 36, 44 increases by exactly 8\displaystyle 8 each time, which mathematically proves that the missing term is 35\displaystyle 35.

*Note on Answer Key Discrepancy:* The textbook answer key lists **Option D** (56\displaystyle 56) as correct. However, 56\displaystyle 56 does not follow any logical, consistent mathematical progression with the rest of the terms. The value 35\displaystyle 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.

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