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Question 57
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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