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A = 1/2^2 + 1/3^2 +.. + 1/8^2 < 1/1.2 + 1/2.3 +... + 1/7.8 = 1 - 1/2 + 1/2 -1/3 +... + 1/7 - 1/8
= 1 - 1/8 < 1
\(\Rightarrowđpcm\)
\(tíchnhaminhftchlaij\)
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Ta thấy:\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}
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\(A=\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{99^2}+\dfrac{1}{100^2}\)
\(=1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{99^2}+\dfrac{1}{100^2}\)
\(\Rightarrow A< 1.\left(\dfrac{1}{2.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{98.99}+\dfrac{1}{99.100}\right)\)
\(\Rightarrow A< 1+\left(\dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{98}-\dfrac{1}{99}+\dfrac{1}{99}-\dfrac{1}{100}\right)\)
\(\Rightarrow A< 1+\left(\dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{100}\right)\)
Mà ta thấy \(\dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{100}< \dfrac{1}{4}+\dfrac{1}{2}=\dfrac{3}{4}\)
\(\Rightarrow A< 1+\dfrac{3}{4}=\dfrac{7}{4}\)
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Lời giải:
$A=1+2+2^2+2^3+2^4$
$2A=2+2^2+2^3+2^4+2^5$
$\Rightarrow A=2A-A=2^5-1$
$\Rightarrow A=2^5-1=32-1=31$
Ta có:
2a = 2 + 2^2 + .... + 2^5
2a - a = 2+ 2^2 + 2^5 - 1 - 2 - 2^2 - .... - 2^4
a = 2^5 - 1 = 32 - 1 = 31