What should come in place of the question mark (?) in the given series? 32,…

In a number series where the plain, first-order differences between consecutive terms are not constant, check whether those first differences themselves form a simple arithmetic progression — that is, whether the second-order differences (the differences between the differences) are constant. When they are, the series has quadratic growth: the next first difference equals the last first difference plus that constant second-order difference, and the next term equals the last term plus this next first difference.

1. Compute the first differences of the given series 32, 37, 47, 62, 82: 37-32=5, 47-37=10, 62-47=15, 82-62=20.

2. Compute the second differences of these values: 10-5=5, 15-10=5, 20-15=5 — a constant value of 5.

3. Since the second difference is constant at 5, the next first difference continues the same pattern: 20+5=25.

4. Add this next difference to the last given term: 82+25=107.

• The pattern gives a closed form t(n) = 32 + 5×(n-1)×n/2 for the nth term. Checking n=5: 32 + 5×4×5/2 = 32+50 = 82, matching the fifth given term.

• Checking n=6 with the same formula: 32 + 5×5×6/2 = 32+75 = 107, the same value obtained by the difference method.

The value that replaces the question mark is 107.