Dacă schițăm un desen orientativ, vom constata că:
[tex]\it AB_1=\dfrac{1}{2},\ \ AB_2=\dfrac{1}{4}=\dfrac{1}{2^2},\ \ \ AB_3=\dfrac{1}{2^3} ,\ ...\ AB_n=\dfrac{1}{2^n}[/tex]
[tex]\it a)\ \ AB_n=\dfrac{1}{1024} \Rightarrow AB_n=\dfrac{1}{2^{10}} \Rightarrow n=10[/tex]
b)
[tex]\it \dfrac{..}{..}\ \ \ \ \dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\ ...\ +\dfrac{1}{2^n}=\dfrac{1}{2}\cdot\dfrac{\Big(\dfrac{1}{2}\Big)^n-1}{\dfrac{1}{2}-1} =\dfrac{1}{2}\cdot\dfrac{1-\dfrac{1}{2^n}}{1-\dfrac{1}{2}}=\\ \\ \\ =\dfrac{1}{2}\cdot\dfrac{\dfrac{2^n-1}{2^n}}{\dfrac{1}{2}}=\dfrac{1}{2}\cdot2\cdot\dfrac{2^n-1}{2^n}=\dfrac{2^n-1}{2^n}[/tex]
