Question

[Ngọc Ý Nhi Trần]

Tính S = 1 + 4 + 4² + 4³ + ...+4²⁰²¹
               _____________________

                             1-4²⁰²²

     ^{Giúp em với ạ!!!}

Answer 1

[Nguyễn Thị Thương Hoài]

S = \(\dfrac{1+4+4^2+4^3+4^4+...+4^{2021}}{1-4^{2022}}\)

Đặt tử số là A  thì S = \(\dfrac{A}{1-4^{2022}}\)

  A =      1 + 4  + 4² + 4³ + 4⁴ + ... + 4²⁰²¹

4A=       4  + 4² + 4³ + 4⁴ + 4⁵ + ....+ 4²⁰²²

4A - A = 4 + 4²+4³+4⁴+4⁵+...+4²⁰²²- (1+4+4²+4³+4⁴+...+4²⁰²¹)

4A     = 4 + 4² + 4³ + 4⁴+4⁵+4²⁰²² - 1 - 4 - 4² - 4³ - 4⁴ - ... - 4²⁰²¹

3A = (4 - 4) +(4² - 4²) + (4³-4³) + (4⁴ -4⁴) +...+(4²⁰²¹- 4²⁰²¹)+4²⁰²²- 1

3A = 4²⁰²² - 1

 A  = \(\dfrac{4^{2022}-1}{3}\)

S  = \(\dfrac{4^{2022}-1}{3}\). \(\dfrac{1}{1-4^{2022}}\)

S = - \(\dfrac{1}{3}\)

Answer 2

[HT.Phong (9A5)]

Ta đặt: \(A=1+4+4^2+...+4^{2021}\)

\(4A=4+4^2+4^3+...+4^{2022}\)

\(4A-A=4+4^2+4^3+...+4^{2022}-1-4-4^2-...-4^{2021}\)

\(3A=4^{2022}-1\)

\(A=\dfrac{4^{2022}-1}{3}\)

\(\Rightarrow S=\dfrac{1+4+4^2+...+4^{2021}}{1-4^{2022}}\)

\(=\dfrac{\left(4^{2022}-1\right):3}{1-4^{2022}}\)

\(=\dfrac{\left(4^{2022}-1\right)\cdot\dfrac{1}{3}}{-\left(4^{2022}-1\right)}\)

\(=-\dfrac{1}{3}\)