What will come in the place of question (?) mark in the following number…

Concept

In a number series, look at the pattern formed by the gaps between consecutive terms. When the gaps themselves are not constant, examine the differences of those gaps (the second differences). A recognisable rule there — such as the second differences growing by a fixed amount — lets you extend the series reliably.

Application

Write the consecutive differences of the given terms 32, 40, 53, 76, ?, 172:

1. 40 − 32 = 8

2. 53 − 40 = 13

3. 76 − 53 = 23

Now take the differences of these gaps (the second differences): 13 − 8 = 5 and 23 − 13 = 10. They increase by a fixed step of 5, so the next second differences are 15 and 20.

1. Next gap = 23 + 15 = 38, so the missing term = 76 + 38 = 114.

2. Following gap = 38 + 20 = 58, giving 114 + 58 = 172.

Cross-check

Rebuilding the whole series gives 32, 40, 53, 76, 114, 172, and the final term lands exactly on 172 as required. This confirms the missing value is 114.