Let [f(x) = 2x^n + , R,n N], and f(4) = 133,f(5) = 255.

Question

Let $$f(x) = 2{x^n} + \lambda ,\lambda  \in R,n \in N$$, and $f(4) = 133,f(5) = 255$. Then the sum of all the positive integer divisors of $(f(3) - f(2))$ is

Accepted Answer

$f(x)=2 x^{n}+\lambda, \lambda \in \mathbb{R}, n \in \mathbb{N}$

$f(4)=2 \cdot 4^{n}+\lambda=133, f(5)=2 \cdot 5^{n}+\lambda=255$

$f(5)-f(4)=2 \cdot\left(5^{n} \cdot 4^{n}ight)=122 \Rightarrow n=3$

$\Rightarrow f(3)-f(2)=2 \cdot\left(3^{n} \cdot 2^{n}ight)=2 \cdot\left(3^{3}-2^{3}ight)=2 	imes 19$

Required sum $=1+2+19+38=60$

Solutions · 1