Question

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer 2

A=20+21+22+23+...++23+...+250250

2�=2+22+23+...+2512A=2+22+23+...+251

2�−�=�=251−202A−A=A=251−20

�=5+52+53+...+599+5100B=5+52+53+...+599+5100

5�=52+53+54+...+5100+51015B=52+53+54+...+5100+5101

5�−�=4�=5101−55B−B=4B=5101−5

�=5101−54B=45101−5​

�=3−32+33−34+...+C=3−32+33−34+...+32007−32008+32009−3201032007−32008+32009−32010

3�=32−33+34−35+...−32008+32009−32010+320113C=32−33+34−35+...−32008+32009−32010+32011

3�+�=4�=32011+33C+C=4C=32011+3

�=32011+34C=432011+3​

�100=5+5×9+5×92+5×93+...+5×999S100​=5+5×9+5×92+5×93+...+5×999

�100=5×(1+9+92+93+...+999)S100​=5×(1+9+92+93+...+999)

9�100=5×(9+92+93+...+999+9100)9S100​=5×(9+92+93+...+999+9100)

9�100−�100=8�100=5×(9100−1)9S100​−S100​=8S100​=5×(9100−1)

�100=5×(9100−1)8S100​=85×(9100−1)​