Thu gọn: S = 1 + x + x^2 + x^3 + …. + x^n

Câu 9: Thu gọn: S = 1 + x + x² + x³ + …. + xⁿ

Trả lời

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

⇒ $S=\frac{x^{n+1}-1}{x-1}$