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chứng minh rằng : 5 ⁶ < 2 ¹⁴ < 5 ⁸

5⁶ = 5.5.5.5.5.5 = 15625

2 ¹⁴= 2.2.2.2.2.2.2.2.2.2.2.2.2.2=16384

5 ^{8 =}5.5.5.5.5.5.5.5=390625

mà 390625 > 16384 > 15625  = > 5 ⁶ < 2 ¹⁴ < 5 ⁸

#HỌCTỐT
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ta có :

5⁶ = 15625

2¹⁴ = 16384

5⁸ =390625

vì 15625 < 16384 < 390625 => 5⁶ < 2¹⁴ < 5⁸

A=\(\frac{10^8+2}{10^8-1}=1+\frac{3}{10^8-1}\)

\(B=\frac{10^8}{10^8-3}=1+\frac{3}{10^8-3}\)

Vì\(10^8-1>10^8-3\)

\(\Rightarrow\frac{3}{10^8-1}< \frac{3}{10^8-3}\)

\(\Rightarrow1+\frac{3}{10^8-1}< 1+\frac{3}{10^8-3}\)

Vậy \(A< B\)

CMR A = 1/5 + 1/6 + 1/7 + ... + 1/17 < 2

A = 1/5 + 1/6 + 1/7 + ... + 1/17

Vì 1/6 < 1/7 < 1/8 < 1/9 < 1/5 và 1/10 < 1/11 < 1/12 < 1/13 < 1/14 <1/15 < 1/16 < 1/17 < 1/8 nên:

A = (1/5 + 1/6 + 1/7 + 1/8 + 1/9) + (1/10 + 1/11 + 1/12 + 1/13 + 1/14+ 1/15 + 1/16 + 1/17)

A < (1/5 + 1/5 + 1/5 + 1/5 + 1/5) + (1/8 + 1/8 + 1/8 + 1/8 + 1/8+1/8 + 1/8 + 1/8)

A < 1 + 1

A < 2

Vậy: A < 2 (đpcm)

a)Ta có:3²⁰=(3²)¹⁰=9¹⁰

             2³⁰=(2³)¹⁰=8¹⁰

       Vì 8¹⁰<9¹⁰

               Suy ra:2³⁰<3²⁰

Gọi dãy trên là A, Ta có: 

1/5²+1/6²+1/7²+...+1/100² < 1/4.5+1/5.6+1/6.7+...+1/99.100

<=> 1/5²+1/6²+1/7²+...+1/100² < 1/4 - 1/100

<=> 1/5²+1/6²+1/7²+...+1/100² < 6/25

Mà 6/25 < 1/4 => A < 1/4

6/25 > 1/6 => A > 1/6

V ậ y: 1/6 < A < 1/4