The-value-of-2-C-0-2-2-2-C-1-2-3-3-C-2-2-4-4-C-3-2-11-11-C-10-is-

Question Number 56140 by gunawan last updated on 11/Mar/19

The value of

2 C_{0} + \frac{2^{2}}{2} C_{1} + \frac{2^{3}}{3} C_{2} + \frac{2^{4}}{4} C_{3} + \dots + \frac{2^{11}}{11} C_{10} is

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

{(1 + x)}^{10} = c_{0} x^{0} + c_{1} x + c_{2} x^{2} + \dots + c_{10} x^{10}

\int {(1 + x)}^{10} d x = \int (c_{0} x^{0} + c_{1} x^{} + c_{2} x^{2} + . . + c_{10} x^{10}) d x

\frac{{(1 + x)}^{11}}{11} = c_{0} \times \frac{x^{1}}{1} + c_{1} \times \frac{x^{2}}{2} + c_{2} \times \frac{x^{3}}{3} + \dots + c_{10} \times \frac{x^{11}}{11} + C

p u t x = 0 b o t h s i d e t o f i n d c o n s t a n t v a l u e o f C

\frac{1}{11} = C

n o w p u t b o t h s i d e x = 2

\frac{{(1 + 2)}^{11}}{11} = c_{0} \times \frac{2^{1}}{1} + c_{1} \times \frac{2^{2}}{2} + c_{2} \times \frac{2^{3}}{3} + \dots + c_{10} \times \frac{2^{11}}{11} + \frac{1}{11}

s o r e q u i r e d a n s w e r i s

\frac{3^{11}}{11} - \frac{1}{11} = c_{0} \times \frac{2^{1}}{1} + c_{1} \times \frac{2^{2}}{2} + c_{2} \times \frac{2^{3}}{3} + \dots + c_{10} \times \frac{2^{11}}{11}

\frac{3^{11} - 1}{11} \leftarrow i t i s t h e a n s w e r