Find the generating function for the sequence $\left \{ a_n \right \} where $
$a_n = \Large \binom{10}{n+1} $ for n = 0,1,2,….
Sol.
$\Large \binom{10}{1} + \binom{10}{2}x + \binom{10}{3}x^2 + \binom{10}{4} x^3 + .... + \binom{10}{10}x^{9}$
multiplying and dividing above equation by x,
$\Large \frac{1}{x} (\binom{10}{1}x + \binom{10}{2}x^2 + \binom{10}{3}x^3 + \binom{10}{4} x^4 + .... + \binom{10}{10}x^{10})$
adding and subtracting $\large \frac{1}{x}$,
$\Large \frac{1}{x} – \frac{1}{x} + \frac{1}{x} (\binom{10}{1}x + \binom{10}{2}x^2 + \binom{10}{3}x^3 + \binom{10}{4} x^4 + .... + \binom{10}{10}x^{10})$
$\Large – \frac{1}{x} + \frac{1}{x} (1 + \binom{10}{1}x + \binom{10}{2}x^2 + \binom{10}{3}x^3 + \binom{10}{4} x^4 + .... + \binom{10}{10}x^{10})$
Using binomial theorem,
$\Large – \frac{1}{x} + \frac{1}{x} ( 1+x)^{10}$
$\Large \color{red}{ \frac{( 1+x )^{10} – 1}{x} }$
Please verify