Home
Class 12
MATHS
Prove that (.^(2n)C(0))^(2) - (.^(2n)C(1...

Prove that `(.^(2n)C_(0))^(2) - (.^(2n)C_(1))^(2) + (.^(2n)C_(2))^(2) - …. + (.^(2n)C_(2n))^(2) = (-1)^(n) .^(2n)C_(n)`

Text Solution

Verified by Experts

`(1+x)^2n(1-(1)/(x))^2n`
`=(.^2nC_0 -(.^2n C_1)x+(.^2nC_2)x62+.....+(.^2nC_2n)x^2n]`
`xx[.^2nC_0 -(.^2nC_1)(1)/(X)+(.^2n C_2)(1)/(x^2)+.....+(.^2n C_2n)(1)/(x^2n)]`
Independent terms of x on RHS
`=(.^2n C_0)^2-(.^2n C_1)^2+(.^2nC_2)^2 -......+(.^2n C_2n)^2`.
LHS `=(1+x)^2n((x-1)/(x))^2n =(1)/(x^2n)(1-x^2)^2n`.
Independent term of x on the LHS `=(-1)^n .^2n C_n`.
Promotional Banner

Similar Questions

Explore conceptually related problems

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

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

Prove that .^(n)C_(0) - ^(n)C_(1) + .^(n)C_(2)- ^(n)C_(3) + "…" + (-1)^(r) + .^(n)C_(r) + "…" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2n+1) 2^(n) .

Prove that .^(n)C_(1) + 2 .^(n)C_(2) + 3 .^(n)C_(3) + "…." + n . ^(n)C_(n) = n 2^(n-1) .

Find the sum 1.^(n)C_(0) + 3 .^(n)C_(1) + 5.^(n)C_(2) + "….." + (2n+1).^(n)C_(n) .

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

If n = 5, then (""^(n)C_(0))^(2)+(""^(n)C_(1))^(2)+(""^(n)C_(2))^(2)+......+(""^(n)C_(5))^(2) is equal to

Prove that .^(2n)C_(n) = ( 2^(n) xx 1 xx 3 xx …(2n-1))/(n!)