If-n-is-an-integer-greater-than-unity-then-the-value-of-a-n-C-1-a-1-n-C-2-a-2-n-C-3-a-3-1-n-a-n-is-

Question Number 53141 by gunawan last updated on 18/Jan/19

If n is an integer greater than unity, then

the value of

a -^{n} C_{1} (a - 1) +^{n} C_{2} (a - 2) -^{n} C_{3} (a - 3) + \dots

. . + {(- 1)}^{n} (a - n) is

Answered by tanmay.chaudhury50@gmail.com last updated on 18/Jan/19

a (1 - {n c}_{1} + {n c}_{2} - {n c}_{3} + \dots + {(- 1)}^{n} {n c}_{n}) + (0 + {n c}_{1} - 2 {n c}_{2} + 3 {n c}_{3} - {n c}_{4} + . . - {(- 1)}^{n} n)

{a s}_{1} + s_{2}

{(1 + x)}^{n} = 1 + {n c}_{1} x + {n c}_{2} x^{2} + {n c}_{3} x^{3} + \dots + {n c}_{n} x^{n}

p u t x = - 1

0 = 1 - {n c}_{1} + {n c}_{2} - {n c}_{3} + . . + {(- 1)}^{n} {n c}_{n}

s o s_{1} = 0

n {(1 + x)}^{n - 1} = 0 + {n c}_{1} + 2 {n c}_{2} x + 3 {n c}_{3} x^{2} + \dots + n \times {n c}_{n} x^{n - 1}

p u t x = - 1

n \times 0 = 0 + {n c}_{1} - 2 {n c}_{2} + 3 {n c}_{3} - \dots + {(- 1)}^{n} \times n \times {n c}_{n}

s o s_{2} = 0

h e n c e {a s}_{1} + s_{2}

= a \times 0 + 0 = 0