Find the next term of the following number series : 2, 3, 10, 15, 26, ?

In some number series, alternate terms follow two different rules based on their position in the series — for example, the square of the position number adjusted up or down by one, with the sign flipping from term to term. Checking a series against each term's position, not just the gaps between consecutive terms, can reveal such an alternating pattern even when the simple differences look irregular.

Number each term by its position n = 1, 2, 3, and so on, and check whether n^2 plus or minus 1 reproduces the series:

1. Position 1: 12 + 1 = 2, which matches the first term.

2. Position 2: 22 - 1 = 3, which matches the second term.

3. Position 3: 32 + 1 = 10, which matches the third term.

4. Position 4: 42 - 1 = 15, which matches the fourth term.

5. Position 5: 52 + 1 = 26, which matches the fifth term.

6. Position 6: 62 - 1 = 35, giving the required next term.

As a cross-check, look at the first differences of the series — 1, 7, 5, 11 — and then at the differences between those differences: 7 - 1 = 6, 5 - 7 = -2, 11 - 5 = 6. These second-level differences alternate between +6 and -2, so the next one should be -2, making the next first difference 11 + (-2) = 9. Adding this to the fifth term gives 26 + 9 = 35, matching the value found above.

So the sixth term of this series is 35.