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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 ________

Given that, $f (x+2) = f(x) + f(x+1) \quad \longrightarrow (1)$

And, $f(11) = 91 , f(15) =617$

Let, $f(10)$ be $k.$

Now, from equation $(1)$

$f(12) = f(10) + f(11)$

$\Rightarrow f(12) = k + 91 \quad \longrightarrow (2)$

$\Rightarrow f(13) = f (11) + f (12)$

$\Rightarrow f(13) = 91 + k + 91 = k + 182 \quad \longrightarrow (3)$

$\Rightarrow f(14) = f(12) +f(13)$

$\Rightarrow f(14) =k + 91 + k + 182$

$\Rightarrow f(14) = 2k + 273 \quad \longrightarrow (4)$

$\Rightarrow f(15) = f(13) + f(14)$

$\Rightarrow 617 = k + 182 + 2k + 273$

$\Rightarrow 3k = 617 – 455$

$\Rightarrow 3k = 162$

$\Rightarrow k = \frac{162}{3}$

$\Rightarrow \boxed{k = 54}$

$\therefore \boxed {f(10) = k = 54}$

Correct Answer $: 54$
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