نویسنده : امیر انصاری

1. $$
   5+5\bigl( \frac{2}{3} \bigr)+5\bigl( \frac{2}{3} \bigr)^2+5\bigl( \frac{2}{3} \bigr)^3+\text{...}
   $$
   
2. $$
   1+\bigl( -\frac{1}{4} \bigr)+\bigl( -\frac{1}{4} \bigr)^2+\bigl( -\frac{1}{4} \bigr)^3+\text{...}
   $$
   
3. $$
   7+7\bigl( \frac{1}{2} \bigr)+7\bigl( \frac{1}{2} \bigr)^2+7\bigl( \frac{1}{2} \bigr)^3+\text{...}
   $$
   

پاسخ

1. $$
   t_1=5\\
   r=\frac{2}{3}\\
   S_{\infty} = \frac{t_1}{1-r}, -1 \lt r \lt 1\\
   S_{\infty} =\frac{5}{1-\frac{2}{3}}=\frac{5}{\frac{1}{3}}=(5)(\frac{3}{1})=15
   $$
   
2. $$
   t_1=1\\
   r=-\frac{1}{4}\\
   S_{\infty} = \frac{t_1}{1-r}, -1 \lt r \lt 1\\
   S_{\infty} = \frac{1}{1-\bigl( -\frac{1}{4} \bigr)} = \frac{1}{\frac{5}{4}}=\frac{4}{5}
   $$
   
3. $$
   t_1=7\\
   r=\frac{1}{2}\\
   S_{\infty} = \frac{t_1}{1-r}, -1 \lt r \lt 1\\
   S_{\infty} =\frac{7}{1-\frac{1}{2}}=\frac{7}{\frac{1}{2}}=14
   $$