Given series is sigma i=1 to infinity 12*[tex]\frac{3}{5} ^{i-1}[/tex],
= [tex]12+ 12*\frac{3}{5} + 12*\frac{3}{5}^{2}+.......[/tex]
Clearly it is a geometric series and it converges if and only if r<1
So, common ratio = [tex]\frac{3}{5}[/tex] < 1
Hence the series is convergent.
Formula for sum of infinite geometric series = [tex]\frac{a}{1-r}[/tex]
Where a is the first term and r is the common ratio.
So, sum of series = [tex]\frac{12}{1-\frac{3}{5} } = \frac{12}{\frac{2}{5} }[/tex]
For taking 2/5 to numerator we have to multiply with reciprocal 5/2 on both numerator and denominator.
Hence sum of series = 12*[tex]\frac{5}{2}[/tex] = 30
