Question

the sum of an infinite geometric series is 162 and the sum of its first n terms is 160. if the inverse of its common rstio is an integer find all possible values of the common ratio n and first term

Answer 1

r= 4, 2, 1
a=10, 144, 160

Answer 2

All possible values of the common ratio n and first term are:

(1) First term = 108, common ratio = 1/3, n = 4

(2) First term = 144, common ratio = 1/9, n = 2

(3) First term = 160, common ratio = 1/81, n = 1

Step-by-step explanation:

Let the first term of the geometric series be a and the common ratio be r

Sum of an infinite geometric series is given by

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

According to the question

$162=\frac{a}{1-r}$

Sum of n terms is given by

$S_n=\frac{a(1-r^n)}{1-r}$

$\implies 160 = \frac{a(1-r^n)}{1-r}$

$\implies 162(1-r^n)=160$

$\implies 1-r^n=\frac{160}{162}$

$\implies r^n=1-\frac{162}{160}$

$\implies r^n=\frac{2}{162}$

$\implies r^n=\frac{1}{81}$

Since the reciprocal of the common ratio is an integer

Therefore, for possible values of n

$\implies r^n=(\frac{1}{3})^4$

This gives, $r=1/3$ and $n=4$

And $a=162(1-1/3)=162\times\frac{2}{3}=108$

Also, $r^n=(\frac{1}{9})^2$

This gives, $r=1/9$ and $n=2$

And $a=162(1-1/9)=162\times\frac{8}{9}=144$

And, $r^n=(\frac{1}{81})^1$

This gives, $r=1/81$ and $n=1$

And $a=162(1-1/81)=162\times\frac{80}{81}=160$

Hope this answer is helpful.

Know More:

Q: If the common ratio of a g.p. is 2/3 and sum to infinity is 12.find first term

Click Here: https://brainly.in/question/12623858

Q: The first term of a gp is 2 and the sum to infinity is 6. find the common ratio.

Click Here: https://brainly.in/question/3158909