Toán (nâng cao)

M

manhnguyen0164

Anh chém tý :D:D:D

a) Ta có $A=1+(\dfrac{1}{2^1}+\dfrac{1}{3})+(\dfrac{1}{2^2}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7})+(\dfrac{1}{2^3}+...+\dfrac{1}{15})+...+(\dfrac{1}{2^{99}}+...+\dfrac{1}{2^{100}-1})$
$A<1+2.\dfrac{1}{2}+2^2.\dfrac{1}{2^2}+...+2^{99}.\dfrac{1}{2^{99}}$
$=\underset{100 số 1}{\underbrace{1+1+...+1+1}}=100$
b) Ta có: $A=1+\dfrac{1}{2}+(\dfrac{1}{3}+\dfrac{1}{2^2})+( \dfrac{1}{5}+...+\dfrac{1}{2^3})+...+(\dfrac{1}{2^{99}+1}+...+\dfrac{1}{2^{100}-1}+\dfrac{1}{2^{100}})-\dfrac{1}{2^{100}}$
$A>1+\dfrac{1}{2}+2.\dfrac{1}{2^2}+2^2.\dfrac{1}{2^3}+...+2^{99}.\dfrac{1}{2^{100}}-\dfrac{1}{2^{100}}$
$=1+\underset{100 số}{\underbrace{\dfrac{1}{2}+...+\dfrac{1}{2}}}-\dfrac{1}{2^{100}}=51-\dfrac{1}{2^{100}}>50$
 
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