Number Series, Coding, Odd Man OutMTP June 2023 Series IQuestion 2069 of 217
All Questions

Find next term of the series 10,69,236,595,?\displaystyle 10, 69, 236, 595, ?

Options

A1254\displaystyle 1254
B1020\displaystyle 1020
C1320\displaystyle 1320
D1200\displaystyle 1200
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Correct Answer

✅ Option a — 1254\displaystyle 1254

All Options:

  • A1254\displaystyle 1254
  • B1020\displaystyle 1020
  • C1320\displaystyle 1320
  • D1200\displaystyle 1200

Detailed Solution & Explanation

To find the next term of the series 10,69,236,595,?\displaystyle 10, 69, 236, 595, ?, we analyze the structure of each term using a quadratic relation:
Tn=Bn2+AnT_n = B_n^2 + A_n
where Bn\displaystyle B_n is a sequence of bases and An\displaystyle A_n is a sequence of addends:

1. **Analyzing the Bases (Bn\displaystyle B_n):**
- B1=3\displaystyle B_1 = 3
- B2=8\displaystyle B_2 = 8 (difference of 8−3=5\displaystyle 8 - 3 = 5)
- B3=15\displaystyle B_3 = 15 (difference of 15−8=7\displaystyle 15 - 8 = 7)
- B4=24\displaystyle B_4 = 24 (difference of 24−15=9\displaystyle 24 - 15 = 9)
We observe the differences between bases are consecutive odd numbers starting from 5\displaystyle 5 (5,7,9\displaystyle 5, 7, 9). Therefore, the next difference must be +11\displaystyle +11:
B5=24+11=35B_5 = 24 + 11 = 35

2. **Analyzing the Addends (An\displaystyle A_n):**
- A1=1\displaystyle A_1 = 1
- A2=5\displaystyle A_2 = 5 (difference of 5−1=4\displaystyle 5 - 1 = 4)
- A3=11\displaystyle A_3 = 11 (difference of 11−5=6\displaystyle 11 - 5 = 6)
- A4=19\displaystyle A_4 = 19 (difference of 19−11=8\displaystyle 19 - 11 = 8)
We observe the differences between addends are consecutive even numbers starting from 4\displaystyle 4 (4,6,8\displaystyle 4, 6, 8). Therefore, the next difference must be +10\displaystyle +10:
A5=19+10=29A_5 = 19 + 10 = 29

3. **Computing each term:**
- T1=32+1=9+1=10\displaystyle T_1 = 3^2 + 1 = 9 + 1 = 10
- T2=82+5=64+5=69\displaystyle T_2 = 8^2 + 5 = 64 + 5 = 69
- T3=152+11=225+11=236\displaystyle T_3 = 15^2 + 11 = 225 + 11 = 236
- T4=242+19=576+19=595\displaystyle T_4 = 24^2 + 19 = 576 + 19 = 595
- T5=352+29=1225+29=1254\displaystyle T_5 = 35^2 + 29 = 1225 + 29 = 1254

This mathematical logic is robust and shows that the next term in the series is **1254\displaystyle 1254**, which corresponds to **Option A**.

*Note on Answer Key Discrepancy:* The textbook/exam answer key lists **Option D** (1200\displaystyle 1200) as correct. However, 1200\displaystyle 1200 does not follow any logical mathematical derivation, whereas 1254\displaystyle 1254 is perfectly and uniquely determined by the quadratic progression above. This represents a typographical error in the official answer key, and the correct option is **Option A**.

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

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