Question

[Nguyễn Hà Chi]

a) Rút gọn tổng A = 2⁰ + 2¹ + 2² + 2³ + .... 2⁵⁰

b) Rút gọn tổng B = 5 + 5² + 5³ +..... +5⁹⁹+ 5¹⁰⁰

c) Rút gọn tổng C = 3 - 3² + 3³ - 3⁴ .....+ 3²⁰⁰⁷ - 3²⁰⁰⁸ + 3²⁰⁰⁹ - 3²⁰¹⁰ 

d) Rút gọn tổng S₁₀₀ = 5 + 5 x 9 + 5 x 9² + 5 x 9³+ ......5 x 9⁹⁹

Answer 1

[Ran Mori]

\(A=2^0+2^1+2^2\)\(+2^3+...+\)\(2^{50}\)

\(2A=2+2^2+2^3+...+2^{51}\)

\(2A-A=A=2^{51}-2^0\)

\(B=5+5^2+5^3+...+5^{99}+5^{100}\)

\(5B=5^2+5^3+5^4+...+5^{100}+5^{101}\)

\(5B-B=4B=5^{101}-5\)

\(B=\frac{5^{101}-5}{4}\)

\(C=3-3^2+3^3-3^4+...+\)\(3^{2007}-3^{2008}+3^{2009}-3^{2010}\)

\(3C=3^2-3^3+3^4-3^5+...-3^{2008}+3^{2009}-3^{2010}+3^{2011}\)

\(3C+C=4C=3^{2011}+3\)

\(C=\frac{3^{2011}+3}{4}\)

\(S_{100}=5+5\times9+5\times9^2+5\times9^3+...+5\times9^{99}\)

\(S_{100}=5\times\left(1+9+9^2+9^3+...+9^{99}\right)\)

\(9S_{100}=5\times\left(9+9^2+9^3+...+9^{99}+9^{100}\right)\)

\(9S_{100}-S_{100}=8S_{100}=5\times\left(9^{100}-1\right)\)

\(S_{100}=\frac{5\times\left(9^{100}-1\right)}{8}\)

Answer 2

[Trần Nhật Minh]

$A=2^0+2^1+2^2$$+2^3+...+$$2^{50}$

$2A=2+2^2+2^3+...+2^{51}$

$2A-A=A=2^{51}-2^0$

$B=5+5^2+5^3+...+5^{99}+5^{100}$

$5B=5^2+5^3+5^4+...+5^{100}+5^{101}$

$5B-B=4B=5^{101}-5$

$B=\genfrac{}{}{0ex}{}{5^{101}-5}{4}$

$C=3-3^2+3^3-3^4+...+$$3^{2007}-3^{2008}+3^{2009}-3^{2010}$

$3C=3^2-3^3+3^4-3^5+...-3^{2008}+3^{2009}-3^{2010}+3^{2011}$

$3C+C=4C=3^{2011}+3$

$C=\genfrac{}{}{0ex}{}{3^{2011}+3}{4}$

$S^{100}=5+5\times 9+5\times 9^2+5\times 9^3+...+5\times 9^{99}$

$S^{100}=5\times \left(1+9+9^2+9^3+...+9^{99}\right)$

$9S^{100}=5\times \left(9+9^2+9^3+...+9^{99}+9^{100}\right)$

$9S^{100}-S^{100}=8S^{100}=5\times \left(9^{100}-1\right)$

$S^{100}=\genfrac{}{}{0ex}{}{5\times \left(9^{100}-1\right)}{8}$