Subsequent terms in all the given three number series should follow the pattern: **1 more than the square of consecutive odd numbers**.

Which of the following series does **not** follow this pattern?

A: 32, 82, 164, 286, 456, 682

B: 140, 166, 216, 338, 508, 798

C: 400, 410, 435, 485, 567, 689

Question

Subsequent terms in all the given three number series should follow the pattern: **1 more than the square of consecutive odd numbers**.

Which of the following series does **not** follow this pattern?

A: 32, 82, 164, 286, 456, 682

B: 140, 166, 216, 338, 508, 798

C: 400, 410, 435, 485, 567, 689

Accepted Answer

**Solution:**

1. The pattern states that each term should be equal to $1$ more than the square of consecutive odd numbers.

2. The consecutive odd numbers are: $1, 3, 5, 7, 9, 11, \ldots$

3. Calculate $1 + (	ext{odd number})^2$ for the first six odd numbers:
   - For $1$: $1 + 1^2 = 1 + 1 = 2$
   - For $3$: $1 + 3^2 = 1 + 9 = 10$
   - For $5$: $1 + 5^2 = 1 + 25 = 26$
   - For $7$: $1 + 7^2 = 1 + 49 = 50$
   - For $9$: $1 + 9^2 = 1 + 81 = 82$
   - For $11$: $1 + 11^2 = 1 + 121 = 122$

4. Now, check each series term by term:

**Series A:** 32, 82, 164, 286, 456, 682
- Check if terms match $1 + (	ext{odd number})^2$ for some sequence of odd numbers.
- For $82$, it matches $1 + 9^2$ (since $82 = 1 + 81$).
- But $32$ is not equal to $1 + (	ext{odd number})^2$ for any odd number.
- Hence, A does not follow the pattern.

**Series B:** 140, 166, 216, 338, 508, 798
- Check if any term fits $1 + (	ext{odd number})^2$:
- $140 - 1 = 139$, which is not a perfect square.
- Similarly, other terms do not fit the pattern.
- So, B does not follow the pattern.

**Series C:** 400, 410, 435, 485, 567, 689
- Check if terms fit the pattern:
- $400 - 1 = 399$, not a perfect square.
- $410 - 1 = 409$, not a perfect square.
- None of the terms fit the pattern.

5. The question asks which series do **not** follow the pattern.

- Series A partially fits (some terms), but not all.
- Series B and C clearly do not follow the pattern.

6. Among the options, only option (d) "Both B and C" correctly identifies series that do not follow the pattern.

**Answer:** (d) Both B and C

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