If ' (n ) ' is a positive integer, then ( _ r=1 ^n r^2 C_r=( ) 2^ n-2 )

Step-by-step solution

( aligned & _ r=1 ^n r^2 C_r=1^2 C₁+2^2 C₂+3^2 C₃+ +n^2 C_n & (1+x)^n=C₀+C₁ x+C₂ x^2+C₃ x^3+ . .+C_n x^n aligned ) On differentiating both sides w.r.t. (x ), we get (n(1+x)^ n-1 =1 C₁+2 C₂ x+3 C₃ x^2+ .+n C_n x^ n-1 ) Now, on multiplying by (x ) both sides, we get (n x(1+x)^ n-1 =1 . C₁ x+2 C₂ x^2+3 C₃ x^3+ +n C_n x^n ) Now again on differentiating both sides w.r.t (x ), we get ( aligned & n [(1+x)^ n-1 +(n-1) x(1+x)^ n-2 ] & =1^2 C₁+2^2 C₂ x+3^2 C₃ x^2+ +n^2 C_n x^ n-1 aligned ) Put (x=1 ), we get ( aligned & 1^2