The sum of the series $^{2020} C_{0} -^{2020} C_{1} +^{2020} C_{2} -^{2020} C_{3} + . . . +^{2020} C_{1010}$ is

Open in App

Solution

${(1 + x)}^{2020} = {^{2020} C}_{0} {(1)}^{2020} {(x)}^{0} + {^{2020} C}_{1} {(1)}^{2019} {(x)}^{^{'}} + . . . . + {^{2020} C}_{2020} {(1)}^{0} x^{2020} P u t x = 1 0 = {^{2020} C}_{0} {(- 1)}^{0} + {^{2020} C}_{1} {(- 1)}^{1} + . . . . + {^{2020} C}_{2020} 0 = {^{2020} C}_{0} - {^{2020} C}_{1} + {^{2020} C}_{2} + . . . . + {^{2020} C}_{2020} N o w, {^{2020} C}_{0} = {^{2020} C}_{2020} (∵ {^{n} C}_{r} = {^{n} C}_{n - r}) S i m i l a r l y {^{2020} C}_{1} = {^{2020} C}_{2010} . . . {^{2020} C}_{1009} = {^{2020} C}_{1011} ∴ 0 = 2 [{^{2020} C}_{0} - {^{2020} C}_{1} + . . . . + {^{2020} C}_{1008} - {^{2020} C}_{1009}] + {^{2020} C}_{1010} ∴ 2 [{^{2020} C}_{0} - {^{2020} C}_{1} + . . . + {^{2020} C}_{1010}] = {^{2020} C}_{1010} ∴ {^{2020} C}_{0} - {^{2020} C}_{1} + {^{2020} C}_{2} + . . . + {^{2020} C}_{1010} = \frac{{^{2020} C}_{1010}}{2} \frac{{^{2020} C}_{1010}}{2} a n s w e r$

Suggest Corrections

Similar questions

3

5

\sum_{r = 0}^{10}

\infty \sum n = 0 r^{n} = S

| r | < 1

\infty \sum n = 0 r^{2 n}

9, 3, 1, \frac{1}{3}, . . . . .

Join BYJU'S Learning Program