Thu gọn: S = 1 + x + x^2 + x^3 + + x^n
Giải thích
S = 1 + x + x2 + x3 + …. + xn
⇔ S = x0 + x1 + x2 + x3 + ...... + xn
⇒ xS = x1 + x2 + x3 + ...... + xn+1
⇒ xS – S = x1 + x2 + x3 + ...... + xn+1 - (x0 + x1 + x2 + x3 + ...... + xn)
⇔ S(x - 1) = x1 + x2 + x3 + ...... + xn+1 - x0 - x1 - x2 - x3 - …... - xn
⇔ S(x - 1) = xn+1 - x0
⇔ S(x-1) = xn+1 - 1
⇒\[S = \frac{{{x^{n + 1}} - 1}}{{x - 1}}\]