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

• A. 59
• B. 60
• C. 61
• D. 58

Answer 1

Answer: (B)

$ f( x )=2 x ^{ n }+\lambda$
$f(4)=133$
$ f(5)=255 $
$ 133=2 \times 4^{ n }+\lambda ...$(1)
$ 255=2 \times 5^{ n }+\lambda....$(2)
$ (2)-(1) $
$ 122=2\left(5^{ n }-4^{ n }\right) $
$ \Rightarrow 5^{ n }-4^{ n }=61 $
$ \therefore n =3 \& \lambda=5 $
Now, $ f(3)-f(2)=2\left(3^3-2^3\right)=38$
Number of Divisors is $1, 2, 19, 38 $; & their sum is $60$