What is the missing number in the pattern given below ? 1, 6, 15, ____, 45,…

Concept: When the first differences between consecutive terms of a series themselves increase by a constant amount, the series has a constant second difference – the missing term is found by extending that steady progression of gaps, not by guessing a fresh pattern from scratch.

Application:

1. Label the terms by position: T1=1, T2=6, T3=15, T4=missing, T5=45, T6=66, T7=91.

2. Compute the gaps that are already known: 6−1=5, 15−6=9, 66−45=21, 91−66=25.

3. These known gaps 5, 9, …, 21, 25 rise by a constant 4 each time (9−5=4 and 25−21=4), so the two missing gaps in between must continue that same run: 13, then 17.

4. Add the first of those two gaps to the term right before the blank: 15+13=28.

Cross-check: Continuing the same run of gaps confirms every later term – 28+17=45, 45+21=66, 66+25=91 – all match the given series exactly, so the missing term is verified independently rather than assumed.