If-1-x-n-C-0-C-1-x-C-2-x-2-C-n-x-n-then-for-n-odd-C-0-2-C-1-2-C-2-2-C-3-2-1-n-C-n-2-is-equal-to-

Question Number 56141 by gunawan last updated on 11/Mar/19

If {(1 + x)}^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n}, then

for n odd, C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - C_{3}^{2} + \dots + {(- 1)}^{n} C_{n}^{2}

is equal to

Answered by tanmay.chaudhury50@gmail.com last updated on 11/Mar/19

{(1 + x)}^{n} = c_{0} + c_{1} x + c_{2} x^{2} + \dots + c_{n} x^{n}

{(1 - \frac{1}{x})}^{n} = c_{0} - c_{1} \times \frac{1}{x} + c_{2} \times \frac{1}{x^{2}} + \dots + {(- 1)}^{n} \times \frac{1}{x^{n}}

{(1 + x) (1 - \frac{1}{x})}^{n}

= {1 - \frac{1}{x} + x - 1}^{n}

= {(x - \frac{1}{x})}^{n}

l e t r + 1 t h t e r m c o n t a i n s x^{0} [x i n d e p e n d e n t t e r m]

{n c}_{r} {(x)}^{n - r} {(\frac{- 1}{x})}^{r}

\frac{n!}{r! (n - r)!} \times x^{n - r} \times {(- 1)}^{r} \times \frac{1}{x^{r}}

\frac{n!}{r! (n - r)!} \times {(- 1)}^{r} \times x^{n - 2 r}

s o n - 2 r = 0 \to r = \frac{n}{2}

h e n c e

c_{0}^{2} - c_{1}^{2} + c_{2}^{2} \dots . + {(- 1)}^{n} c_{n}^{2} = \frac{n!}{(\frac{n}{2})! (\frac{n}{2})!} \times {(- 1)}^{\frac{n}{2}}