Question

Tính nhanh tổng : 1+1/4+1/4²+1/4³+....+1/4²⁰⁰⁰

Answer 1

Đặt A=1+1/4+1/4²+1/4³+....+1/4²⁰⁰⁰

\(4A=4\left(1+\frac{1}{4}+...+\frac{1}{4^{2000}}\right)\)

\(4A=4+1+...+\frac{1}{4^{1999}}\)

\(4A-A=\left(4+1+...+\frac{1}{4^{1999}}\right)-\left(1+\frac{1}{4}+...+\frac{1}{4^{2000}}\right)\)

\(3A=4-\frac{1}{4^{2000}}\)

\(A=\frac{4-\frac{1}{4^{2000}}}{3}\)

Answer 2

Đặt tổng trên là A

Ta có: A=1+1/4+1/4²+1/4³+...+1/4²⁰⁰⁰ (1)

         1/4.A= 1/4+1/4²+1/4³+1/4⁴+...+1/4²⁰⁰¹(2)

Lấy (2) trừ (1), ta có:

-3/4.A=1/4²⁰⁰¹-1

A=(1/4²⁰⁰¹-1):(-3/4)

Answer 3

\(\text{Đặt }\)\(A=1+\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+.....+\frac{1}{4^{2000}}\)

\(Khi\text{đ}\text{ó}4\text{A}=4+1+\frac{1}{4}+.....+\frac{1}{4^{2001}}\)

\(\Rightarrow4A-A=4+\frac{1}{4^{2001}}-\frac{1}{4^{2000}}\)

Answer 4

dap số là (5/16)^2020