522 1235 2661 4800 7652 11217 ?

Concept

When the first differences between consecutive terms are not constant, check whether the differences themselves grow in a fixed pattern — for example, as consecutive multiples of one number k (k, 2k, 3k, 4k, ...). Then the difference-of-differences (the second difference) is constant, and the next difference is simply the next multiple of k.

Application

Compute the consecutive differences of the given series:

1. 1235 − 522 = 713

2. 2661 − 1235 = 1426

3. 4800 − 2661 = 2139

4. 7652 − 4800 = 2852

5. 11217 − 7652 = 3565

6. Dividing each difference by 713 gives 1, 2, 3, 4, 5 — the differences are consecutive multiples of 713.

7. The next difference must therefore be the sixth multiple: 6 × 713 = 4278.

8. Adding this to the last term: 11217 + 4278 = 15495.

Cross-check

The second differences between consecutive first differences are all equal to 713 (1426−713=713, 2139−1426=713, 2852−2139=713, 3565−2852=713), confirming a constant second difference. Extending it once more, 4278−3565=713, which independently confirms the sixth difference is 4278 and the missing term is 15495.