Question

Rút gọn A= 2^100-2^99+2^98-2^97+...+2^2-2

Answer 1

Ta có: A = 2¹⁰⁰ - 2⁹⁹ + 2⁹⁸ - 2⁹⁷ + ... + 2² - 2 (gồm 100 hạng tử)

A = (2¹⁰⁰ - 2⁹⁹) + (2⁹⁸ - 2⁹⁷) + ... + (2² - 2) (gồm 50 cặp)

A = 2⁹⁹(2 - 1) + 2⁹⁷.(2 - 1) + ... + 2(2 - 1)

A = 2⁹⁹ + 2⁹⁷ + .... + 2

2²A = 2²(2⁹⁹ + 2⁹⁷ + ... + 2)

4A = 2¹⁰¹ + 2⁹⁹ + ... + 2³

4A - A = (2¹⁰¹ + 2⁹⁹ + ... + 2³) - (2⁹⁹ + 2⁹⁷  + ... + 2)

3A = 2¹⁰¹ - 2

A = \(\frac{2^{101}-2}{3}\)

Answer 2

\(A=2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)

\(2A=2^{201}-2^{100}+2^{99}-2^{98}+...+2^3-2^2\)

\(2A+A=2^{101}-2^{100}+2^{99}-2^{98}+...+2^3-2^2+2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)

\(3A=2^{201}-2\)

\(A=\frac{2^{201}-2}{3}\)