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\(5S=1+\frac{2}{5}+\frac{3}{5^2}+...+\frac{2015}{5^{2014}}\Rightarrow4S=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2014}}-\frac{2015}{5^{2015}}\)
Đặt B = \(1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2014}}\)
=> 5B = \(5+1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2013}}\)
=> 4B = \(5-\frac{1}{5^{2014}}
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Ta có :S=1/11+1/12+1/13+...+1/20<1/10+1/10+1/10+...+1/10(20 số hạng 1/10)
=>S<1/10.20=1/2<5/6
ĐPCM
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A = 1 + 2 + 22 + 23 + .... +298
2A = 2(1 + 2 + 22 + 23 +....+298)
2A = 2 + 22 + 23 + 24 +....+299
2A - A = (2 + 22 + 23 + 24 +.....+ 299) - (1 + 2 + 22 + 23 +.....+298)
A = ( 2 - 2 ) + ( 22 - 22 ) + (23 - 23).....+(298 - 298) + 299 - 1
A = 0 + 0 + 0 +.....+0 + 299 - 1
A = 299 - 1
So sánh :
299 - 1 và 5 . 298
299 - 1 < 5 . 298
( Biết hay sai thì chịu nhe hehe phần so sánh tui
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Ta có: \(S=1+3+3^2+...+3^{20}\)
\(\Rightarrow3S=3+3^2+3^3+...+3^{21}\)
\(\Rightarrow3S-S=\left(3+3^2+3^3+...+3^{21}\right)-\left(1+3+3^2+...+3^{20}\right)\)
\(\Rightarrow2S=3^{21}-1\)
\(\Rightarrow S=\left(3^{21}-1\right).\frac{1}{2}\)
\(\Rightarrow S=3^{21}.\frac{1}{2}-\frac{1}{2}\)
Vì \(3^{21}.\frac{1}{2}-\frac{1}{2}< 3^{21}.\frac{1}{2}\) nên \(A< \frac{1}{2}.3^{21}\)
Vậy \(A< \frac{1}{2}.3^{21}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
S = 1 + 2 + 2 ^ 2 + 2 ^ 3 + ... + 2 ^ 9
2S = 2 + 2 ^ 2 + 2 ^ 3 + 2 ^ 4 + .... + 2 ^ 10
2S - S = ( 2 + 2 ^ 2 + 2 ^ 3 + 2 ^ 4 + .... + 2 ^ 10 )
- ( 1 + 2 + 2 ^ 2 + 2 ^ 3 + ... + 2 ^ 9 )
S = 2 ^ 10 - 1
S = 2 ^ 8 . 2 ^ 2 - 1
S = 2 ^ 8 . 4 - 1
S < 2 ^ 8 . 1 < 5 . 2 ^ 8
Vậy S < 5 . 2 ^ 8
nhanh đi