Question

1+2x+3x^2+4x^3+...+nx^(n-1)

Answer 1

Let S = 1+2x+3x^2+4x^3+.............nx^(n-1) ... (1)

Multiply both sides by 'x'

xS = x + 2x^2+ 3x^3+.............nx^(n) ... (2)

Subtract (2) from (1)

S (1- x) = 1 + x + x^2 + ..... + x^(n-1) + nx^(n)

Except the last term, the rest is Geometric Progression with first term '1' and ratio 'x' with total 'n' terms

S(1-x) = 1 (1-x)/(1-x^n) + n X^n

S = 1/(1 - x^n) + nx^n/(1-x)