Question

[Nguyễn Quỳnh Anh]

Tính hợp lí:

1+2+2²+2³+2⁴+.......+2⁹⁹+2¹⁰⁰

Answer 1

[l҉o҉n҉g҉ d҉z҉]

Đặt A = 1 + 2 + 2² + 2³ + 2⁴ + ... + 2⁹⁹ + 2¹⁰⁰

=> 2A = 2( 1 + 2 + 2² + 2³ + 2⁴ + ... + 2⁹⁹ + 2¹⁰⁰ )

            = 2 + 2² + 2³ + 2⁴ + ... + 2¹⁰⁰ + 2¹⁰¹

=> A = 2A - A

          = 2 + 2² + 2³ + 2⁴ + ... + 2¹⁰⁰ + 2¹⁰¹ - ( 1 + 2 + 2² + 2³ + 2⁴ + ... + 2⁹⁹ + 2¹⁰⁰ )

          = 2 + 2² + 2³ + 2⁴ + ... + 2¹⁰⁰ + 2¹⁰¹ - 1 - 2 - 2² - 2³ - 2⁴ - ... - 2⁹⁹ - 2¹⁰⁰ 

          = 2¹⁰¹ - 1

Answer 2

[๓เภђ ภوยץễภ ђảเ]

Đătj S= 1+2+2²+2³+2⁴+.......+2⁹⁹+2¹⁰⁰

\(\Rightarrow2S=2+2^2+2^3+...+2^{101}\)

\(\Rightarrow2S-S=(2+2^2+2^3+...+2^{101})-\)\((1+2+2^2+...+2^{100})\)

\(\Rightarrow S=2^{101}-1\)