Question 21

# If $$f(x + 2) = f(x) + f(x + 1)$$ for all positive integers x, and $$f(11) = 91, f(15) = 617$$, then $$f(10)$$ equals

Solution

$$f(x + 2) = f(x) + f(x + 1)$$
As we can see, the value of a term is the sum of the 2 terms preceding it.

It has been given that $$f(11) = 91$$ and $$f(15) = 617$$.
We have to find the value of $$f(10)$$.

Let $$f(10)$$ = b
$$f(12)$$ = b + 91
$$f(13)$$ = 91 + b + 91 = 182 + b
$$f(14)$$ = 182+b+91+b = 273+2b
$$f(15)$$ = 273+2b+182+b = 455+3b
It has been given that 455+3b = 617
3b = 162
=> b = 54

Therefore, 54 is the correct answer.