Number Series, Coding, Odd Man OutMTP Sep 24 Series IIQuestion 2081 of 217
All Questions

Find next term of the series 1,3,14,30,55,91,?\displaystyle 1, 3, 14, 30, 55, 91, ?

Options

A130\displaystyle 130
B140\displaystyle 140
C150\displaystyle 150
D160\displaystyle 160
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Correct Answer

✅ Option b — 140\displaystyle 140

All Options:

  • A130\displaystyle 130
  • B140\displaystyle 140
  • C150\displaystyle 150
  • D160\displaystyle 160

Detailed Solution & Explanation

To find the next term of the series 1,3,14,30,55,91,?\displaystyle 1, 3, 14, 30, 55, 91, ?, we must first identify a significant typographical error in the question statement as printed in the textbook/MTP:

The second term, **3**, is a typographical error for **5**.

With this correction, the true logical series is:
1,5,14,30,55,91,?1, 5, 14, 30, 55, 91, ?
Let us analyze this sequence, which represents the **square pyramidal numbers** (the sum of the squares of the first n\displaystyle n consecutive integers):

Let Tn\displaystyle T_n represent the n\displaystyle n-th term of the series:
- 1st term: T1=12=1\displaystyle T_1 = 1^2 = 1
- 2nd term: T2=12+22=5\displaystyle T_2 = 1^2 + 2^2 = 5 (printed as 3\displaystyle 3 due to a typo)
- 3rd term: T3=12+22+32=14\displaystyle T_3 = 1^2 + 2^2 + 3^2 = 14
- 4th term: T4=12+22+32+42=30\displaystyle T_4 = 1^2 + 2^2 + 3^2 + 4^2 = 30
- 5th term: T5=12+22+32+42+52=55\displaystyle T_5 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55
- 6th term: T6=12+22+32+42+52+62=91\displaystyle T_6 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91

**Alternative Method (Differences):**
We can analyze the differences between consecutive terms:
- T2−T1=5−1=4=22\displaystyle T_2 - T_1 = 5 - 1 = 4 = 2^2
- T3−T2=14−5=9=32\displaystyle T_3 - T_2 = 14 - 5 = 9 = 3^2
- T4−T3=30−14=16=42\displaystyle T_4 - T_3 = 30 - 14 = 16 = 4^2
- T5−T4=55−30=25=52\displaystyle T_5 - T_4 = 55 - 30 = 25 = 5^2
- T6−T5=91−55=36=62\displaystyle T_6 - T_5 = 91 - 55 = 36 = 6^2
The differences are perfect squares of consecutive integers (22,32,42,52,62\displaystyle 2^2, 3^2, 4^2, 5^2, 6^2).
Following this logical pattern, the next difference must be 72=49\displaystyle 7^2 = 49. We add 49\displaystyle 49 to the 6th term (91\displaystyle 91):
T7=91+49=140T_7 = 91 + 49 = 140

This mathematical proof uniquely demonstrates that the next term in the series is **140\displaystyle 140**, which corresponds to **Option B**.

*Note on Answer Key Discrepancy:* The official answer key lists **Option C** (150\displaystyle 150) as correct. This is a double error: first, the typo of 3\displaystyle 3 instead of 5\displaystyle 5 in the question, and second, the selection of 150\displaystyle 150 in the answer key (which has no mathematical justification). The mathematically correct answer is **Option B** (140\displaystyle 140).

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

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