Answer:
[tex]8 \ terms[/tex]
Step-by-step explanation:
Let total required terms[tex]=n[/tex]
[tex]t_1=125, \ t_2=25, \ t_3=5, \ t_4=1\\\frac{t_2}{t_1}=\frac{t_3}{t_2}=\frac{t_4}{t_3}=\frac{1}{5} \\[/tex]
So it is a geometric series with constant ratio [tex]\frac{1}{5}[/tex].
Sum of [tex]n[/tex] terms of geometric series[tex]=\frac{a(1-r^n)}{1-r}[/tex]
Where [tex]a[/tex] is first term and [tex]r[/tex] is constant ratio.
[tex]Here \ a=125 \ and \ r=\frac{1}{5} \\\\\frac{125(1-\frac{1}{5}^n) }{1-\frac{1}{5} } =\frac{97656}{625}\\\\ 125(1-\frac{1}{5}^n)= \frac{97656}{625}\times\frac{4}{5} \\\\(1-\frac{1}{5}^n)=\frac{390624}{625\times5\times125} \\\\(1-\frac{1}{5}^n)=\frac{390624}{390625} \\\\\frac{1}{5}^n=1-\frac{390624}{390625}\\\\\frac{1}{5}^n=\frac{1}{390625} \\\\\frac{1}{5}^n=\frac{1}{5}^8 \ \ \ \ \ \ \ \ \ \ \ \ \ (as\ 5^8=390625)\\\\Compare \ both \ side\\\\n=8[/tex]
