Home
Class 12
MATHS
Prove that (2nC0)^2+(2nC1)^2+(2nC2)^2-+(...

Prove that `(2nC_0)^2+(2nC_1)^2+(2nC_2)^2-+(2nC_(2n))^2=(-1)^n2nC_ndot`

Text Solution

Verified by Experts

`underset(r=0)overset(2n)sum(-1)^(r)(.^(2n)C_(r))^(2)=underset(r=0)overset(2n)sum(-1)^(r).^(2n)C_(r).^(2n)C_(r)`
`= underset(r=0)overset(2n)sum(-1)^(r) .^(2n)C_(r).^(2n)C_(2n-r)`
= Coefficient of `x^(2n)` in `(1-x)^(2n)(1+x)^(2n)`
= Coefficient of `x^(2n)` is `(1-x)^(2n)`
`= (-1)^(n).^(2n)C_(n)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.8|10 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise (Single)|90 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.6|10 Videos

  • AREA UNDER CURVES

    CENGAGE|Exercise Question Bank|10 Videos

  • CIRCLE

    CENGAGE|Exercise MATRIX MATCH TYPE|6 Videos

Similar Questions

Explore conceptually related problems

Prove that ^n C_0^n C_0-^(n+1)C_1^n C_1+^(n+2)C_2^n C_2-=(-1)^ndot

Prove that ^n C_0 ^(2n)C_n- ^n C_1 ^(2n-2)C_n+ ^n C_2^(2n-4)C_n-=2^ndot

Prove that C_(0)^(2) + C_(1)^(2) + C_(2)^(2) +…+C_(n)^(2) = (2n!)/(n!)^(2)

Prove that "^n C_0^(2n)C_n-^n C_1^(2n-1)C_n+^n C_2xx^(2n-2)C_n++(-1)^n^n C_n^n C_n=1.

Given that C_1+2C_2x+3C_3x^2++2nC_(2n)x^(2n-1)=2n(1+x)^(2n-1), w h e r eC_r=(2n)!//[r !(2n-r)!]; r=0,1,2, ,2n , then prove that C1 2-2C2 2+3C3 2--2n C2n2=(-1)^nn C_ndot

Prove that ^n C_1(^n C_2)(^n C_3)^3(^n C_n)^nlt=((2^n)/(n+1))^(n+1_C()_2),AAn in Ndot

If n is a positive integer, prove that 1-2n+(2n(2n-1))/(2!)-(2n(2n-1)(2n-2))/(3!)++(-1)^(n-1)(2n(2n-1)(n+2))/((n-1)!)= (-1)^(n+1)(2n)!//2(n !)^2dot

Prove that 1^(2)+2^(2)+3^(2)+.....+n^(2)=(n(n+1)(2n+1))/6

Prove that (C_1)/1-(C_2)/2+(C_3)/3-(C_4)/4++((-1)^(n-1))/n C_n=1+1/2+1/3++1/ndot