Question

[Scorpion]

Cho S = 1 - 2 + 2^2 - 2^3 + .... - 2^2005 + 2^2006

a) Tính 2S; 3S

b) Tính 3S - 2^2007

Answer 1

[Trần Ngoc an]

a) goi E=1-2=-1

\(\Rightarrow N=2^2-2^3+....+2^{2005}+2^{2006}\)

Ta co :

\(N=2^2-2^3+.....+2^{2005}+2^{2006}\)

\(2N=2^3-2^4+2^5-2^6+.....+2^{2005}-2^{2006}+2^{2007}\)

\(N=2^{2007}-2^2\)

=> S=E+N

S=-1+\(2^{2007}-2^2\)

\(\Rightarrow3S=-3+3.2^{2007}-3.2^2=3.2^{2007}-15\)

\(\Rightarrow2S=-2=2^{2008}-2^3=2^{2008}-10\)

\(vay:2S=2^{2008}-10/3S=3.2^{2007}-15\)

con phan b thi bn cho mik ty nhe

Answer 2

[Nguyễn Thị Thùy Trâm]

Cho S = 1 - 2 + 2² - 2³ + .... - 2²⁰⁰⁵ + 2²⁰⁰⁶

a) Tính 2S; 3S

S = (1 - 2) + (2² - 2³ + .... - 2²⁰⁰⁵ + 2²⁰⁰⁶)

Ta gọi M là (1 - 2) = -1

N là (2² - 2³ + .... - 2²⁰⁰⁵ + 2²⁰⁰⁶)

N = 2² - 2³ + .... - 2²⁰⁰⁵ + 2²⁰⁰⁶

2N = 2(2² - 2³ + .... - 2²⁰⁰⁵ + 2²⁰⁰⁶)

2N = 2³ - 2⁴ + .... - 2²⁰⁰⁶ + 2²⁰⁰⁷

2N - N = (2³ - 2⁴ + .... - 2²⁰⁰⁶ + 2²⁰⁰⁷) - (2² - 2³ + .... - 2²⁰⁰⁵ + 2²⁰⁰⁶)

N = 2²⁰⁰⁷ - 2²

Vì S = M + N

Nên S = (-1) + 2²⁰⁰⁷ - 2²

2S = 2[(-1) + 2²⁰⁰⁷ - 2²]

2S = (-2) + 2²⁰⁰⁸ - 2³

2S = 2²⁰⁰⁸ - 10

3S = 3[(-1) + 2²⁰⁰⁷ - 2²]

3S = (-3) + 3. 2²⁰⁰⁷ - 3. 2²

3S = (-3) + 3. 2²⁰⁰⁷ - 3. 4

3S = 3. 2²⁰⁰⁷ - 3. 4 - 3

3S = 3. 2²⁰⁰⁷ - 15

b) Tính 3S - 2²⁰¹⁷

3S = 3. 2²⁰⁰⁷ - 15

3S - 2²⁰⁰⁷ = 3.2²⁰⁰⁷ - 15 - 2²⁰⁰⁷

3S - 2²⁰⁰⁷ = 3.2²⁰⁰⁷ - 2²⁰⁰⁷ - 15

3S - 2²⁰⁰⁷ = 3.2²⁰⁰⁷ - 1.2²⁰⁰⁷ - 15

3S - 2²⁰⁰⁷ = (3 - 1) . 2²⁰⁰⁷ - 15

3S - 2²⁰⁰⁷ = 2 . 2²⁰⁰⁷ - 15

a) ➤ 2S = 2²⁰⁰⁸ - 10

➤ 3S = 3. 2²⁰⁰⁷ - 15

b) ➤ 3S - 2²⁰⁰⁷ = 2 . 2²⁰⁰⁷ - 15