In the series 3, 11, 23, 39, 59, .......... The next term will be

When the differences between consecutive terms of a series are not constant, check whether those differences themselves form an arithmetic progression — that is, whether the difference-of-differences (the second difference) stays constant. A series with a constant second difference is a quadratic-type series, and its next term is found by extending this pattern of differences one more step.

Applying this to the given series:

1. Write down the given terms: 3, 11, 23, 39, 59.

2. Find the first differences between consecutive terms: 11 − 3 = 8, 23 − 11 = 12, 39 − 23 = 16, 59 − 39 = 20.

3. Find the differences between these first differences (the second differences): 12 − 8 = 4, 16 − 12 = 4, 20 − 16 = 4 — constant at 4, confirming this is a quadratic-type series.

4. Extend the pattern: the next first difference must also increase by 4, so it is 20 + 4 = 24.

5. Add this difference to the last given term: 59 + 24 = 83.

Cross-check: fitting the terms to a quadratic confirms this. With a constant second difference of 4, the nth term has the form an = 2n2 + bn + c. Using the first three terms (3, 11, 23) to solve for the coefficients gives an = 2n2 + 2n − 1. Checking this formula against the 4th and 5th terms: a4 = 2(16) + 8 − 1 = 39 and a5 = 2(25) + 10 − 1 = 59 — both match the given series, confirming the formula. The 6th term is then a6 = 2(36) + 12 − 1 = 83, the same value obtained by extending the differences.

So, 83 is the correct next term of the series.