Question

1/5+1/5²+.....1/5^100. mate bac.

Answer 1

   
[tex]\displaystyle\\ \frac{1}{5}+ \frac{1}{5^2}+ \frac{1}{5^3}+\cdots + \frac{1}{5^{100}}=?\\\\ \text{Este o progresie geometrica cu: }\\\\ \begin{cases} b_1 = \dfrac{1}{5}\\\\ q = \dfrac{1}{5}\\\\ \text{Suma are 100 de termeni.} \end{cases}\\\\\\ \text{Folosim formula: }\\\\ S_n=b_1\cdot \frac{q^n-1}{q-1} \\\\\\\\ S_{100}= \frac{1}{5} \cdot \frac{ \left(\dfrac{1}{5}\right)^{100}-1 }{\dfrac{1}{5}-1} =\\\\\\ =\frac{1}{5} \cdot \frac{1- \left(\dfrac{1}{5}\right)^{100} }{1-\dfrac{1}{5}} = [/tex]


[tex]\displaystyle\\ =\frac{1}{5} \cdot \frac{1- \left(\dfrac{1}{5}\right)^{100} }{\dfrac{4}{5}} = \frac{1}{5}\cdot \dfrac{5}{4}\cdot \left( 1- \left(\dfrac{1}{5}\right)^{100} \right)=\\\\\\ =\dfrac{1}{4}\cdot \left( 1- \left(\dfrac{1}{5}\right)^{100} \right)= \dfrac{1}{4}\cdot \left( 1- \dfrac{1}{5^{100}} \right)= \dfrac{1}{4}\cdot \dfrac{5^{100}-1}{5^{100}} =\boxed{\bf\dfrac{5^{100}-1}{4\cdot5^{100}}} [/tex]