Question

So sánh hai số sau:

a/ A=1+5+5\(^2\)+5\(^3\)+...+5\(^{99}\)+5\(^{100}\) và B=\(\dfrac{5^{101}}{4}\)

b/ M=\(\dfrac{9^{20}+2}{9^{19}+2}\) và N=\(\dfrac{9^{19}+3}{9^{18}+3}\)

Answer 1

a)

\(A=1+5+5^2+5^3+................+5^{99}\)

\(\Rightarrow5A=5+5^2+5^3+................+5^{99}+5^{100}\)

\(\Rightarrow5A-A=\left(5+5^2+5^3+.........+5^{99}+5^{100}\right)-\left(1+5+5^2+.......+5^{99}\right)\)

\(\Rightarrow4A=5^{100}-1\)

\(\Rightarrow A=\dfrac{5^{100}-1}{4}\)

Ta có :

\(A=\dfrac{5^{100}-1}{4}< B=\dfrac{5^{100}}{4}\Rightarrow A< B\)

b) Chưa có nghĩ ra!!

Answer 2

a, \(A=1+5+5^2+...+5^{100}\\ =>5A=5+5^2+5^3+...........+5^{101}\\ =>5A-A=\left(5+5^2+5^3+......+5^{101}\right)-\left(1+5+5^2+...5^{100}\right)\\ 4A=5^{101}-1\\ =>A=\dfrac{5^{101}-1}{4}->\left(1\right)\)

Theo đề: \(B=\dfrac{5^{101}}{4}->\left(2\right)\)

Từ (1) và (2), ta thấy: \(\dfrac{5^{101}-1}{4}< \dfrac{5^{101}}{4}\\ =>A< B\)

Answer 3

a) A= 1+5+5²+5³+..........+5⁹⁹+5¹⁰⁰

5A= 5+5²+5³⁺5⁴+..........+5¹⁰⁰+5¹⁰¹

5A-A= (5+5²+5³+5⁴+......+5¹⁰⁰+5¹⁰¹) - (1+5+5²+5³+.....+5⁹⁹+5¹⁰⁰)

4A= 5¹⁰¹-1

A= \(\dfrac{5^{101}-1}{4}\)

Vì A= \(\dfrac{5^{101}-1}{4}\)< B= \(\dfrac{5^{101}}{4}\) nên A < B

Answer 4

a, \(A=1+5+5^2+...+5^{99}+5^{100}\)

\(5A=5\left(1+5+5^2+...+5^{99}+5^{100}\right)\)

\(5A=5+5^2+5^3+...+5^{100}+5^{101}\)

\(4A=5A-A=\left(5+5^2+5^3+...+5^{100}+5^{101}\right)-\left(1+5+5^2+...+5^{99}+5^{100}\right)\) \(4A=5^{101}-1\)

\(A=\dfrac{5^{101}-1}{4}\)

\(A=\dfrac{5^{101}-1}{4}< \dfrac{5^{101}}{4}=B\)

\(\Rightarrow A< B\)