this-question-was-repeatd-six-times-in-a-various-exams-between-1971-to-2001-if-C-0-C-1-C-2-C-n-are-the-coefficients-in-the-expansion-of-1-x-n-then-c-0-2C-1-3C-2-n-1-C-n-

Question Number 106690 by M±th+et+s last updated on 06/Aug/20

t h i s q u e s t i o n w a s r e p e a t d s i x t i m e s

i n a v a r i o u s e x a m s b e t w e e n 1971 t o 2001 .

i f C_{0}, C_{1}, C_{2} \dots \dots ., C_{n} a r e t h e c o e f f i c i e n t s

i n t h e e x p a n s i o n o f {(1 + x)}^{n} t h e n

c_{0} + 2 C_{1} + 3 C_{2} \dots \dots . . (n + 1) C_{n} = ?

Answered by prakash jain last updated on 06/Aug/20

(r + 1) C_{r} = {r C}_{r} + C_{r}

{(1 + x)}^{n} = \underset{j = 0}{\sum^{n}} C_{j} x^{j}

d i f f e r e n t i a t e

n {(1 + x)}^{n - 1} = \underset{j = 0}{\sum^{n}} {j C}_{j} x^{j - 1}

x = 1

n 2^{n - 1} = \underset{j = 0}{\sum^{n}} {j C}_{j}

\underset{r = 0}{\sum^{n}} (r + 1) C_{r} = \underset{r = 0}{\sum^{n}} {r C}_{r} + \underset{r = 0}{\sum^{n}} C_{r}

= n 2^{n - 1} + 2^{n}

Commented by M±th+et+s last updated on 06/Aug/20

w e l l d o n e s i r