Question

determine the 18th term of the GP whose 5th term is 1 and common ratio is 2/3​

Answer 1

The 18th term of the GP is $a_{18}=\frac{2^{13}}{3^{13}}$.

Step-by-step explanation:

It is given that 18th term of the GP whose 5th term is 1 and common ratio is 2/3​.

nth term of a GP is

$a_n=ar^{n-1}$

where, a is first term and r is common ratio.

5th term of GP is

$a_5=a(\frac{2}{3})^{5-1}$

$1=a(\frac{2}{3})^{4}$

$(\frac{3}{2})^4=a$

First term of GP is $a=(\frac{3}{2})^4$.

18th term of GP is

$a_{18}=(\frac{3}{2})^4(\frac{2}{3})^{18-1}$

$a_{18}=(\frac{3}{2})^4(\frac{2}{3})^{17}$

Using the properties of exponents we get

$a_{18}=\frac{3^4}{2^4}\times \frac{2^{17}}{3^{17}}$

$a_{18}=\frac{2^{17-4}}{3^{17-4}}$

$a_{18}=\frac{2^{13}}{3^{13}}$

Therefore, 18th term of the GP is $a_{18}=\frac{2^{13}}{3^{13}}$.

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