Find the next term in the number series given below: 1, 7, 25, 61, ?

Concept

A number series fixes each term by a rule that depends on the term’s position n. When the first differences between consecutive terms are not constant, the two standard routes are to look for a pattern in those differences, or to fit a closed form an = f(n).

Either route is only admissible once it reproduces every term already supplied; a rule that matches only the last two terms is not established, and agreement between two independent routes is the strongest confirmation available inside the item itself.

Application

1. Index the given terms by position: a1 = 1, a2 = 7, a3 = 25, a4 = 61.

2. Take the first differences: 7 − 1 = 6, 25 − 7 = 18, 61 − 25 = 36. These are not constant, so the difference pattern itself must be identified.

3. Compare them with the triangular numbers Tk = k(k + 1)/2, whose values run 1, 3, 6, 10, … : the differences are 6 × 1, 6 × 3 and 6 × 6, that is 6Tk for k = 1, 2, 3.

4. Extending that pattern, the next difference is 6T4 = 6 × 10 = 60.

5. Hence a5 = a4 + 60 = 61 + 60 = 121.

Cross-check

An independent route is to fit a closed form in n and test it against every supplied term.

• Try an = n3 − n + 1 : 13 − 1 + 1 = 1, 23 − 2 + 1 = 7, 33 − 3 + 1 = 25, 43 − 4 + 1 = 61.

• All four supplied terms are reproduced, so the closed form is admissible: a5 = 53 − 5 + 1 = 125 − 5 + 1 = 121.

• The difference route and the closed-form route agree, so the extension does not depend on a single guessed pattern.

The fifth term of the series is 121.