If $\displaystyle a= \frac{1}{3^{223}}+1$ and $\displaystyle f(n)= \binom{n}{0}a^{n-1}-\binom{n}{1}a^{n-2}+...........+(-1)^{n-1}\binom{n}{n-1}a^{0}$
Then value of $f(2007)+f(2008) = $
$\bf{My\; Try::}$ Multiply both side by $a\;,$ We get
$$af(n) = \underbrace{\binom{n}{0}a^{n}-\binom{n}{1}a^{n-1}+...........+(-1)^{n-1}\binom{n}{n-1}a+(-1)^n}_{\bf{(a-1)^n}}+(-1)^{n+1}$$
So we get $$af(n)=(a-1)^n+(-1)^{n+1}\Rightarrow f(n) = \frac{(a-1)^n+(-1)^{n+1}}{a}$$
So $$f(2007)+f(2008) = \frac{1}{a}\left[(a-1)^{2007}+1+(a-1)^{2008}-1\right]$$
So we get $$f(2007)+f(2008) = \frac{(a-1)^{2007}\cdot a}{a} = (a-1)^{2007}=3^{-223\times 2007}$$
My Solution did not match here.
Where i have done Wrong, Help me
Thanks
