\(A=\frac{1}{7}.\left(\frac{1}{10}-\frac{1}{11}+\frac{1}{11}-\frac{1}{12}+...+\frac{1}{68}-\frac{1}{70}\right)\)

\(A=\frac{1}{7}.\left(\frac{1}{10}-\frac{1}{70}\right)=\frac{1}{7}.\frac{3}{35}=\frac{3}{245}\)

A=\(\frac{7}{10.11}\)+\(\frac{7}{11.12}\)+\(\frac{7}{12.13}\)+...+\(\frac{7}{69.70}\)

A=\(\frac{7}{10}\)-\(\frac{7}{11}\)+\(\frac{7}{11}\)-\(\frac{7}{12}\)+\(\frac{7}{12}\)-\(\frac{7}{13}\)+...+\(\frac{7}{69}\)-\(\frac{7}{70}\)

A=\(\frac{7}{10}-\frac{7}{70}\)

A=\(\frac{7}{10}-\frac{1}{10}\)

Ạ=\(\frac{6}{10}=\frac{3}{5}\).

C=7/10x11+7/11x12+7/12x13+.................+7/69x70

C=1x7/10x11+1x7/11x12+...........+1x7/69x70

C=7(1/10x11+1/11x12+1/12x13+....+1/69x70)

C=7(1/10-1/11+1/11-1/12+1/12-1/13+.......+1/69-1/70)

C=7(1/10-1/70)

C=7(7/70-1/70)

C=7x6/70

C=3/5

\(A=1+7+7^2+7^3+...+7^{2016}\)

\(7A=7\left(1+7+7^2+...+7^{2016}\right)\)

\(7A=7+7^2+7^3+...+7^{2017}\)

\(6A=7A-A=7+7^2+7^3+...+7^{2017}-1-7-7^2-...-7^{2016}\)

\(6A=7^{2017}-7\)

\(A=\left(\frac{7^{2017}-7}{6}\right)\)

\(\left(\frac{7^{2017}-7}{6}\right)\)

\(A=\dfrac{7}{9}\left(9+99+999+...+999...9\right)\)

\(=\dfrac{7}{9}\left(10-1+10^2-1+10^3-1+...+10^{10}-1\right)\)

\(=\dfrac{7}{9}\left(10+10^2+...+10^{10}-10\right)\)

\(=\dfrac{7}{9}\left(10.\dfrac{10^{10}-1}{10-1}-10\right)=\dfrac{7}{9}\left(\dfrac{10^{11}}{9}-\dfrac{10}{9}-10\right)\)

\(=\dfrac{7}{81}.10^{11}-\dfrac{700}{81}\)

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A = 7+7³+7⁵+7⁷+....+7⁹⁹

7²A = 7³+7⁵+7⁷+7⁹+.....+7¹⁰¹

48A = 7²A - A = 7¹⁰¹ - 7

=> A = \(\frac{7^{101}-7}{48}\)

Ta có:A=7+7²+7³+...+7²⁰¹⁰

=>7A=7(7+7²+7³+...+7²⁰¹⁰⁾

7A=49+7³+7⁴+...+7²⁰¹¹

=>7A-A=(49+7³+7⁴+...+7²⁰¹¹)-(7+7²+7³+...+7²⁰¹⁰)

=>6A=7²⁰¹¹-7

=>A=(7²⁰¹¹-7):6

Vậy A=(7²⁰¹¹-7):6

A=7^1+7^2+...7^2010

7×A=7^2+7^3+.....7^2011

7×A-A=(7^2+7^3+...+7^2011)-(7^1+7^2+....+7^2010)

=7^2011-7^1

\(A=\frac{7}{10.11}+\frac{7}{11.12}+\frac{7}{12.13}+...+\frac{7}{69.70}\)

\(\Rightarrow A=7.\left(\frac{1}{10.11}+\frac{1}{11.12}+\frac{1}{12.13}+....+\frac{1}{69.70}\right)\)

\(\Rightarrow A=7.\left(\frac{1}{10}-\frac{1}{11}+\frac{1}{11}-\frac{1}{12}+\frac{1}{12}-\frac{1}{13}+...+\frac{1}{69}-\frac{1}{70}\right)\)

\(\Rightarrow A=7\left(\frac{1}{10}-\frac{1}{70}\right)\)

\(\Rightarrow A=7.\frac{3}{35}\)

\(\Rightarrow A=\frac{3}{5}\)

A=7+7² +7³+...+7^n-1+7ⁿ

7A = 7² + 7³ + 7⁴ + ... + 7ⁿ + 7ⁿ⁺¹

7A - A = ( 7² + 7³ + 7⁴ + ... + 7ⁿ + 7ⁿ⁺¹ ) - ( 7+7² +7³+...+7^n-1+7ⁿ )

6A = 7ⁿ⁺¹ - 7

A = \(\frac{7^{n+1}-7}{6}\)