Question

[Nguyễn ☠Văn☠Xuân☠]

\(A=5+5^2+5^3+5^4+...+5^{2004}\)

\(B=7^1+7^2+7^3+...+7^{2015}\)

\(C=4^5+4^6+4^7+...+4^{2016}\)

Answer 1

[Mashiro Shiina]

\(A=5+5^2+5^3+5^4+...+5^{2004}\)

\(5A=5^2+5^3+5^4+5^5+...+5^{2005}\)

\(5A-A=\left(5^2+5^3+5^4+5^5+...+5^{2005}\right)-\left(5+5^2+5^3+5^4+...+5^{2004}\right)\)

\(4A=5^{2005}-5\)

\(A=\dfrac{5^{2005}-5}{4}\)

\(B=7^1+7^2+7^3+....+7^{2015}\)

\(7B=7^2+7^3+7^4+....+7^{2016}\)

\(7B-B=\left(7^2+7^3+7^4+...+7^{2016}\right)-\left(7+7^2+7^3+....+7^{2015}\right)\)

\(6B=7^{2016}-7\)

\(B=\dfrac{7^{2016}-7}{6}\)

\(C=4^5+4^6+4^7+...+4^{2016}\)

\(4C=4^6+4^7+4^8+...+4^{2017}\)

\(4C-C=\left(4^6+4^7+4^8+...+4^{2017}\right)-\left(4^5+4^6+4^7+...+4^{2016}\right)\)

\(3C=4^{2017}-4^5\)

\(C=\dfrac{4^{2017}-4^5}{3}\)

Answer 2

[Phạm Thị Ngọc Dung]

A = 5 + 5² + 5³ + 5⁴ + ... + 5²⁰⁰⁴

5A = 5² + 5³ + 5⁴ + 5⁵ + ... + 5²⁰⁰⁵

5A - A = 5²⁰⁰⁵ - 5

4A = 5²⁰⁰⁵ - 5

A = (5²⁰⁰⁵ - 5) : 4

B = 7¹ + 7² + 7³ + ... + 7²⁰¹⁵

7B = 7² + 7³ + 7⁴ + ... + 7²⁰¹⁶

7B - B = 7²⁰¹⁶ - 7

6B = 7²⁰¹⁶ - 7

B = (7²⁰¹⁶ - 7) : 6

C = 4⁵ + 4⁶ + 4⁷ + ... + 4²⁰¹⁶

4C = 4⁶ + 4⁷ + 4⁸ + ... + 4²⁰¹⁷

4C - C = 4²⁰¹⁷ - 4⁵

3C = 4²⁰¹⁷ - 4⁵

C = (4²⁰¹⁷ - 4⁵) : 3