IIT JEE , AIEEE Forum - Mathematics , Algebra - Sum of 1 + (1+x) +(1+x+x^2)+(1+x+x^2+x^3)+(1+x+x^2+x^3+x^4).. up to n terms

sriram.a

In simple way take S=n+(n-1)x+(n-2)x^2+.........................+x^n --------------------------------(1)

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

subtract (2)-(1)

S(x-1)={x^(n+1)+x^n+............................+x+1}+(n-1)

$S(x-1)=\frac{x^{n+1}-1}{x-1}+n-1=\frac{x^{n+1}-1+(x-1)n-x+1}{x-1}$

$S=\frac{x^{n-1}+(n-1)x-n}{{(x-1)}^2}$