Polynomial of 11th Degree – Polynomial of 11th Degree

Question

Let $f(x)$ be a polynomial of degree $11$ such that $f(x)=\frac{1}{x+1}$,for $x=0,1,2,3…….,11$.
Then what is the value of $f(12)?$

My attempt at this is:
Let $f(x)=a_0+a_1x+a_2x^2+a_3x^3+……+a_{11}x^{11}$

$f(0)=\frac{1}{0+1}=1=a_0$

$f(1)=\frac{1}{1+1}=\frac{1}{2}=a_0+a_1+a_2+a_3+……+a_{11} $

$f(2)=\frac{1}{2+1}=\frac{1}{3}=a_0+2a_1+4a_2+8a_3+……+2^{11}a_{11} $
.
.
.
$f(11)=\frac{1}{11+1}=\frac{1}{12}=a_0+11a_1+11^2a_2+11^3a_3+……+11^{11}a_{11} $

for calculating $f(12)$, I need to calculate $a_0,a_1,a_2,….,a_11$ but I could solve further.Is my approach right,how can I solve further or there is another right way to solve it.

$(A)\frac{1}{13}$

$(B)\frac{1}{12}$

$(C)0 $

$(D)\frac{1}{7}$

which one is correct answer?

Answer 1

HINT:

Let $(x+1)f(x)=1+A\prod_{r=0}^{11}(x-r)$ where $A$ is an arbitrary constant