Question 8

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

Solution

Expression : $$f(x) = 2x + f(x - 1)$$ and $$f(1) = 1$$

Putting, $$x = 2$$

=> $$f(2) = 2 \times 2 + f(1) = 4 + 1 = 5$$

and $$f(3) = 2 \times 3 + f(2) = 6 + 5 = 11$$

Similarly, $$f(4) = 2 \times 4 + f(3) = 8 + 11 = 19$$

The pattern followed is : $$f(n) = n^2 + (n - 1)$$

Now, $$n = 31$$

$$\therefore f(31) = 31^2 + (31 - 1) = 961 + 30 = 991$$

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