Answer to Question #275282 in Discrete Mathematics for Good heart

Question

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.

Answer 1

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

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.

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