There are four number series given below. One number in each series is incorrect. Identify the wrong number in each series and find the difference between the third term (P) and the sixth term (Q) of series D, which follows the same pattern as series C.

Series A: 24, 31.5, 46.5, 69, 98, 136.5, 181.5

Series B: 5926, 886, 166, 46, 22, 18, 14

Series C: 11, 18, 44, 107, 231, 445, 788

Series D: 23, 30, P, 119, 243, Q, 800

Find the difference between P and Q.

Question

There are four number series given below. One number in each series is incorrect. Identify the wrong number in each series and find the difference between the third term (P) and the sixth term (Q) of series D, which follows the same pattern as series C.

Series A: 24, 31.5, 46.5, 69, 98, 136.5, 181.5

Series B: 5926, 886, 166, 46, 22, 18, 14

Series C: 11, 18, 44, 107, 231, 445, 788

Series D: 23, 30, P, 119, 243, Q, 800

Find the difference between P and Q.

Accepted Answer

**Solution:**

1. **Identify the wrong numbers in the given series:**

- **Series A:** The pattern is an increasing sequence with differences increasing by 7.5, 15, 22.5, 30, 37.5, 45.
  - Correct series should be: 24, 31.5, 46.5, 69, **99**, 136.5, 181.5
  - Wrong number is **98** (should be 99).

- **Series B:** The pattern involves division and subtraction:
  - 5926 to 886 (approx divide by 6.7), 886 to 166 (divide by ~5.3), 166 to 46 (divide by ~3.6), 46 to 22 (divide by 2), then 22 to 16 (not 18), 16 to 14.
  - Wrong number is **18** (should be 16).

- **Series C:** The pattern is based on cubes minus 1:
  - Terms correspond to $2^3 - 1 = 7$, $3^3 - 1 = 26$, $4^3 - 1 = 63$, $5^3 - 1 = 124$, $6^3 - 1 = 215$, $7^3 - 1 = 342$.
  - Given series: 11, 18, 44, 107, 231, 445, 788
  - Wrong number is **445** (should be 342).

2. **Series D follows the same pattern as Series C:**

- First term = 23
- Terms correspond to cubes minus 1 starting from $2^3 - 1$:
  - 1st term: 23 (given)
  - 2nd term: 30
  - 3rd term (P): should be $4^3 - 1 = 63$
  - 4th term: 119
  - 5th term: 243
  - 6th term (Q): should be $7^3 - 1 = 342$
  - 7th term: 800

3. **Calculate the difference between P and Q:**

$$	ext{Difference} = Q - P = 342 - 56 = 402$$

**Answer:** 402

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