Number Series, Coding, Odd Man OutMTP June 2023 Series IIQuestion 2072 of 217
All Questions

105,115.5,150,162.5,203,?\displaystyle 105, 115.5, 150, 162.5, 203, ?

Options

A217\displaystyle 217
B217.5\displaystyle 217.5
C210.5\displaystyle 210.5
DNone of these
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Correct Answer

✅ Option b — 217.5\displaystyle 217.5

All Options:

  • A217\displaystyle 217
  • B217.5\displaystyle 217.5
  • C210.5\displaystyle 210.5
  • DNone of these

Detailed Solution & Explanation

To find the next term of the series 105,115.5,150,162.5,203,?\displaystyle 105, 115.5, 150, 162.5, 203, ?, let us analyze the differences between consecutive terms:

- T2−T1=115.5−105=10.5\displaystyle T_2 - T_1 = 115.5 - 105 = 10.5
- T3−T2=150−115.5=34.5\displaystyle T_3 - T_2 = 150 - 115.5 = 34.5
- T4−T3=162.5−150=12.5\displaystyle T_4 - T_3 = 162.5 - 150 = 12.5
- T5−T4=203−162.5=40.5\displaystyle T_5 - T_4 = 203 - 162.5 = 40.5

We observe that the difference sequence 10.5,34.5,12.5,40.5,…\displaystyle 10.5, 34.5, 12.5, 40.5, \dots consists of two alternating progressions:

1. **Odd-indexed Differences (1st, 3rd, ... differences):**
- 10.5→12.5→…\displaystyle 10.5 \to 12.5 \to \dots
- This progression increases by exactly +2\displaystyle +2 at each step. Therefore, the 5th difference must be:
12.5+2=14.512.5 + 2 = 14.5
2. **Even-indexed Differences (2nd, 4th, ... differences):**
- 34.5→40.5→…\displaystyle 34.5 \to 40.5 \to \dots
- This progression increases by exactly +6\displaystyle +6 at each step.

Using the next odd-indexed difference (14.5\displaystyle 14.5), we calculate the 6th term by adding it to the 5th term (203\displaystyle 203):
T6=203+14.5=217.5T_6 = 203 + 14.5 = 217.5

This mathematical logic uniquely and rigorously proves that the next term is **217.5\displaystyle 217.5**, which corresponds to **Option B**.

*Note on Answer Key Discrepancy:* The textbook/exam answer key lists **Option C** (210.5\displaystyle 210.5) as the correct answer. However, 210.5\displaystyle 210.5 does not fit the mathematical progression of differences, whereas 217.5\displaystyle 217.5 satisfies the alternating arithmetic progression of differences perfectly. This represents a 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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