A = 5 + 5² + 5³ + ... + 5²⁰²³

⇒ 5A = 5² + 5³ + 5⁴ + ... + 5²⁰²⁴

⇒ 4A = 5A - A

= (5² + 5³ + 5⁴ + ... + 5²⁰²⁴) - (5 + 5² + 5³ + ... + 5²⁰²³)

= 5²⁰²⁴ - 5

⇒ A = (5²⁰²⁴ - 5)/4

A = 5 + 5² + 5³ + ... + 5²⁰²³

⇒ 5A = 5² + 5³ + 5⁴ + ... + 5²⁰²⁴

⇒ 4A = 5A - A

= (5² + 5³ + 5⁴ + ... + 5²⁰²⁴) - (5 + 5² + 5³ + ... + 5²⁰²³)

= 5²⁰²⁴ - 5

⇒ A = (5²⁰²⁴ - 5)/4

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A = 5 + 5² + 5³ + ... + 5⁴⁹ + 5⁵⁰

⇒ 5A = 5² + 5³ + 5⁴ + ... + 5⁵⁰ + 5⁵¹

⇒ 4A = 5A - A

= (5² + 5³ + 5⁴ + ... + 5⁵⁰ + 5⁵¹) - (5 + 5² + 5³ + ... + 5⁴⁹ + 5⁵⁰)

= 5⁵¹ - 5

⇒ A = (5⁵¹ - 5) : 4

51. She/ study/ hard/ in order/ pass/exam.
=> She studies hard in order to pass the exam.
52. Mary/ asked/ me/ go/ market/ for/ her.
=> Mary asked me to go to the market for her.
53. Sarah/ have/ live/ Bien Hoa city/ for/ 5 years.
=> Sarah have lived in Bien Hoa city for 5 years.
54. What/ they/ going/ do/ this weekend?
=> Waht are they going to do in this weekend?
55. How long/ have/ you/ study/ English? 
=> How long have you studied English?
56. She/ old/ enough/ put on/ clothes/ herself.
=> She is old enough to put on clothes by herself.
57. There/ used / be/ theatre/ this neighborhood. 
=> There used to be a theatre in this neighboorhood.
58. My mother/ told/ me/ not/ play/ computers/ games/ more/ 2 hours.
=> My mother told me not to play computers games more than 2 hours.
59. My father/ humorous/ and/ enjoy/ tell/ jokes.
=> My father is humorous and he enjoys telling jokes.
60. Minh/ same age/ Hoa.
=> Minh is as the same age as Hoa.

51. She/ study/ hard/ in order/ pass/exam.
=> She studies hard in order to pass the exam.
52. Mary/ asked/ me/ go/ market/ for/ her.
=> Mary asked me to go to the market for her.
53. Sarah/ have/ live/ Bien Hoa city/ for/ 5 years.
=> Sarah has lived in Bien Hoa city for 5 years.
54. What/ they/ going/ do/ this weekend?
=> What are they going to do in this weekend?
55. How long/ have/ you/ study/ English? 
=> How long have you studied English?
56. She/ old/ enough/ put on/ clothes/ herself.
=> She is old enough to put on clothes by herself.
57. There/ used / be/ theatre/ this neighborhood. 
=> There used to be a theatre in this neighboorhood.
58. My mother/ told/ me/ not/ play/ computers/ games/ more/ 2 hours.
=> My mother told me not to play computers games more than 2 hours.
59. My father/ humorous/ and/ enjoy/ tell/ jokes.
=> My father is humorous and he enjoys telling jokes.
60. Minh/ same age/ Hoa.
=> Minh is as the same age as Hoa.

1-2+3-4+5-6+...+51-52+53

=(1-2)+(3-4)+...+(51-52)+53

=(-1)+(-1)+...+(-1)+53

=(-1)×26+53

=-26+53

=27

1-2+3-4+5-6+...+51-52+53

=(1-2)+(3-4)+...+(51-52)+53

=(-1)+(-1)+...+(-1)+53

=(-1)×26+53

=-26+53

=27

a: \(7\cdot\left(-2\right)^3-12\cdot\left(-5\right)+\left(-17\right)\)

\(=7\cdot\left(-8\right)+60-17\)

=-56+43

=-13

b: \(1632-37-\left(-157\right)-163-1532\)

\(=\left(1632-1532\right)-37-163+157\)

=100-200+157

=57

c: \(47\cdot\left(-918\right)+\left(-53\right)\cdot918\)

\(=918\left(-47\right)+\left(-53\right)\cdot918\)

\(=918\cdot\left(-47-53\right)\)

\(=918\left(-100\right)=-91800\)

d: \(\left(-52\right)\cdot\left(-281\right)+\left(-52\right)\cdot181\)

\(=\left(-52\right)\left(-281+181\right)\)

\(=\left(-52\right)\cdot\left(-100\right)=5200\)

a: 

=-56+43

=-13

b: 

=100-200+157

=57

c: 

d: 

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5A=5+5^2+...+5^2023

=>4A=5^2023-1

=>\(A=\dfrac{5^{2023}-1}{4}\)

\(2B-A=\dfrac{5^{2023}}{4}-\dfrac{5^{2023}-1}{4}=\dfrac{1}{4}\)

a) \(S=5+5^2+...+5^{2006}\)

\(5S=5^2+5^3+...+5^{2007}\)

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

\(4S=5^{2007}-5\)

\(S=\dfrac{5^{2007}-5}{4}\)

b) \(S=5+5^2+5^3+...+5^{2006}\)

\(S=\left(5+5^4\right)+\left(5^2+5^5\right)+...+\left(5^{2003}+5^{2006}\right)\)

\(S=5\cdot\left(1+5^3\right)+5^2\cdot\left(1+5^3\right)+...+5^{2003}\cdot\left(1+5^3\right)\)

\(S=\left(1+5^3\right)\cdot\left(5+5^2+...+5^{2003}\right)\)

\(S=126\cdot\left(5+5^2+...+5^{2003}\right)\) ⋮ 126