Question

cho tong S=1+2+2^2+2^3+...+2^99.tim so tu nhien n biet: S+1=4^n+2

Answer 1

S = 1 + 2 + 2² + 2³ + ..... + 2⁹⁹

2S = 2 + 2² + 2³ + ..... + 2⁹⁹ + 2¹⁰⁰

2S - S = (2 + 2² + 2³ + ..... + 2⁹⁹ + 2¹⁰⁰) - (1 + 2 + 2² + 2³ + ..... + 2⁹⁹)

S = 2¹⁰⁰ - 1

S + 1 = 2¹⁰⁰ - 1 + 1 = 4^{n + 2}

=> 2¹⁰⁰ = 4^{n + 2}

=> (2²)⁵⁰ = 4^{n + 2}

=> 4⁵⁰ = 4^{n + 2}

=> n + 2 =50

=> n = 48