Question

Tính hợp lí:

1+2+2²+2³+2⁴+.......+2⁹⁹+2¹⁰⁰

Answer 1

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

Đătj S= 1+2+2²+2³+2⁴+.......+2⁹⁹+2¹⁰⁰

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

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

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

Answer 3

Đặt A = 1 + 2 + 2² + ... + 2⁹⁹ + 2¹⁰⁰

=> 2A = 2 + 2² + 2³ + ... + 2¹⁰⁰ + 2¹⁰¹

Khi đó 2A - A = (2 + 2² + 2³ + ... + 2¹⁰⁰ + 2¹⁰¹) - (1 + 2 + 2² + ... + 2⁹⁹ + 2¹⁰⁰)

                  A = 2¹⁰¹ - 1

Vậy A = 2¹⁰¹ - 1

Answer 4

\(1+2+2^2+2^3+2^4+.....+2^{99}+2^{100}\)

\(=1+2^1+2^2+2^2.2+2^4+2^4.2+....+2^{99}+2^{99}.2\)

\(=\left(1+2^1+2^2.2+2^4.2+....+2^{98}.2+2^{100}\right).\left(2.49\right)\)

\(=1+2^1+2^2+2^4+.....+2^{98}+2^{100}.98.\left(2.49\right)\)

\(=1.2^{51}.98^2\)

\(=2^{108}.34\)

Answer 5

Đặt \(D=1+2+2^2+...+2^{100}\)

=> \(2D=2+2^2+2^3+...+2^{101}\)

=> \(2D-D=\left(2+2^2+...+2^{101}\right)-\left(1+2+...+2^{100}\right)\)

=> \(D=2^{101}-1\)

Vậy \(D=2^{101}-1\)