What is the tenth term of the sequence 1, 5, 14, 30, …?

Concept

When successive increments follow consecutive squares, the nth term is the cumulative sum of the squares from 1 through n.

The governing identity is Σk2 = n(n + 1)(2n + 1) ÷ 6, for k = 1 through n.

Application

1. Compare consecutive terms: 5 − 1 = 4 = 22, 14 − 5 = 9 = 32, and 30 − 14 = 16 = 42.

2. The pattern therefore continues by adding 52, 62, and so on through 102. Equivalently, the tenth term is Σk2 for k = 1 through 10.

3. Using the identity gives 10 × 11 × 21 ÷ 6 = 385.

Cross-check

Starting from the fourth term, add the next six increments: 30 + 25 + 36 + 49 + 64 + 81 + 100 = 385.

Result

The tenth term is 385.