Question

f is a function for which $f(1) = 1$ and $f(x) = 2x + f(x - 1)$ for each natural number $x \ge 2$. Find $f(31)$.

Accepted Answer

D. $f(1) = 1$ $f(2) = 2(2) + 1 = 5$ $f(3) = 2(3) + 5 = 11$ $f(4) = 2(4) + 11 = 19$ In general, $f(x) = f(1) + 2\sum_{k=2}^x k = 1 + 2\left(\frac{x(x+1)}{2} - 1\right) = x^2 + x - 1$. For $x = 31$: $f(31) = 31^2 + 31 - 1 = 961 + 30 = 991$.

Functions: Recurrence Relation

Explanation

Practise this under exam conditions