Find the missing number ‘X’ of the following series: 5, 13, 25, X, 61

Concept

This series is a second-order (quadratic) sequence: the first differences between consecutive terms are not constant, but those first differences themselves increase by a fixed amount at each step (a constant second difference). To find a missing term, compute the known first differences, identify that fixed amount, and extend the pattern.

Application

1. Find the first difference between the 2nd and 1st terms: 13 − 5 = 8.

2. Find the first difference between the 3rd and 2nd terms: 25 − 13 = 12.

3. Compare the two first differences: 12 − 8 = 4, so each first difference is 4 more than the one before it.

4. Apply the same constant increase to get the next first difference: 12 + 4 = 16.

5. Add this difference to the 3rd term to get the missing term: 25 + 16 = 41, so X = 41.

6. Apply the constant increase once more to get the final first difference: 16 + 4 = 20.

Cross-check

Adding the predicted final first difference to X gives 41 + 20 = 61, which matches the last term of the given series. This confirms X = 41 is consistent with the constant-second-difference rule governing the whole series.