S
=
5
2
2
+
5
3
2
+
5
4
2
+
.
.
.
+
5
100
2
.
Chứng tỏ rằng
2
<
S
<
5
S = 5 2 2 + 5 3 2 + 5 4 2 + . . . + 5 100 2 . Chứng tỏ rằng 2 < S < 5
By Everleigh
By Everleigh
S
=
5
2
2
+
5
3
2
+
5
4
2
+
.
.
.
+
5
100
2
.
Chứng tỏ rằng
2
<
S
<
5
Ta có: $\frac{1}{2^2}$ < $\frac{1}{1.2}$ = $\frac{1}{1}$ : $\frac{1}{3^2}$ < $\frac{1}{2.3}$ = $\frac{1}{2}$ – $\frac{1}{3}$ : … : $\frac{1}{100^2}$ < $\frac{1}{99.100}$ = $\frac{1}{99}$ – $\frac{1}{100}$
=> S < 5 ( 1 – $\frac{1}{100}$ < 5.1 = 5 => S<5 )
lại có : $\frac{1}{2^2}$ > $\frac{1}{2.3}$ = $\frac{1}{2}$ – $\frac{1}{3}$ : $\frac{1}{3^2}$ > $\frac{1}{3.4}$ = $\frac{1}{3}$ – $\frac{1}{4}$ : $\frac{1}{100^2}$ > $\frac{1}{100.101}$ = $\frac{1}{100}$ – $\frac{1}{101}$
=> S>5 ( $\frac{1}{2}$ – $\frac{1}{101}$ ) = 5. $\frac{101-2}{2.101}$ = $\frac{5.99}{2.101}$ . 2,45 => S > 2
vậy 2<S <5 => Đpcm
Đáp án+Giải thích các bước giải:
`S=5/2^2+5/3^2+…+5/100^2>5/2.3+5/3.4+…+5/100.101`$\\$`=5.(1/2.3+1/2.3+…+1/100.101)`$\\$`=5.(1/2-1/3+1/3-1/4+…+1/100-1/101)`$\\$`=5.(1/2-1/101)≈2,4>2`
`S=5/2^2+5/3^2+…+5/100^2<5/1.2+5/2.3+…+5/99.100`$\\$`=5.(1/1.2+1/2.3+…+1/99.100)`$\\$`=5.(1-1/2+1/2-1/3+…+1/99-1/100)`$\\$`=5.(1-1/100)≈4,95<5`$\\$`=>2<S<5`