IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

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(1+x)ⁿ = C₀ + C₁x + C₂x² + ....... + C_nxⁿ

Differentiate wrt x :

n(1+x)^n-1 = C₁ + 2C₂x + 3C₃x² + ....... + nC_nx^n-1

Multiply both sides with x :

nx(1+x)^n-1 = C₁x + 2C₂x² + ....... + C_nxⁿ

Again differentiating both sides wrt x :

n(1+x)^n-1 + n(n-1)x(1+x)^n-2 = 1²C₁ + 2²C₂x + 3²C₃x² + ....... + n²C_nx^n-1

Now put x = 1 to get the result :

1²C₁ + 2²C₂ + 3²C₃ + ....... + n²C_n = n(2^n-1) + n(n-1)2^n-2

= (n)(2^n-2)(2 + n - 1)

= (n)(n+1)(2^n-2)

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