Consider the following series 56, 28, 14.. I. Find 17th term. ii. Find the sum of the series if it continues indefinitely iii. Find 20th term.

$\frac{56}{28}=\frac{1}{2}=\frac{14}{28}$

We have an geometric series with $a=56, r=\dfrac{1}{2}.$

Since $|r|=\dfrac{1}{2}<1,$ the geometric series $\displaystyle\sum_{n=0}^{\infin}56(\dfrac{1}{2})^n$ converges.

$a_n=ar^{n-1}$

$S=\dfrac{a}{1-r}$

i.

$a_{17}=56(\dfrac{1}{2})^{17-1}=\dfrac{7}{8192}$

ii.

$S=\dfrac{56}{1-\dfrac{1}{2}}=112$

iii.

$a_{20}=56(\dfrac{1}{2})^{20-1}=\dfrac{7}{65536}$