In the following number series, only one number is incorrect. Select the…

Concept

In a constant-difference (arithmetic) series, every consecutive pair of terms differs by the same fixed step d, so the list of successive differences is itself constant. A single mistyped term therefore disturbs exactly two neighbouring differences: the one entering that term falls short of d and the one leaving it overshoots d by the same amount.

Application

Take the successive differences of the printed series 109, 135, 161, 177, 213, 239, 265:

1. 135 − 109 = 26

2. 161 − 135 = 26

3. 177 − 161 = 16

4. 213 − 177 = 36

5. 239 − 213 = 26

6. 265 − 239 = 26

Four of the six differences equal 26, so the intended step is d = 26. The two irregular differences, 16 and 36, sit next to each other, and the only term they share is the fourth printed term, 177. Their sum is 16 + 36 = 52 = 2 x 26, which is the signature of exactly one mistyped term: replacing that single term restores both differences at once. The value that makes both differences 26 is 161 + 26 = 187.

Cross-check

With 187 in place the series reads 109, 135, 161, 187, 213, 239, 265, and every successive difference is 26. Equivalently the nth term is 109 + 26(n - 1): that formula reproduces 213 = 109 + 26 x 4, 239 = 109 + 26 x 5 and 265 = 109 + 26 x 6 exactly as printed, so no term other than the fourth needs changing.

Hence the number that does not belong in the printed series is 177; it should be 187.