Write a rule for the nth term of the geometric sequence and then find $a}_{5$ given $a}_{4}=\frac{189}{1000},r=\frac{3}{5$

miniliv4
2022-10-03
Answered

Write a rule for the nth term of the geometric sequence and then find $a}_{5$ given $a}_{4}=\frac{189}{1000},r=\frac{3}{5$

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Mario Monroe

Answered 2022-10-04
Author has **12** answers

We know we can write every geometric sequence in the form of below:

$a}_{n}={a}_{1}\times {r}^{n-1$

and for now what we have to do is to figure out what is our first term ( $a}_{1$ ) and we can do it easily cause we have $a}_{4$:

$a}_{4}=\frac{189}{1000}={a}_{1}\times {\left(\frac{3}{5}\right)}^{3$

and we have to solve this equation for $a}_{1$

$a}_{1}=\frac{\frac{189}{1000}}{\frac{{3}^{3}}{{5}^{3}}}=\frac{189\times 125}{1000\times 27}=\frac{7}{8$

Now we can rewrite our equation for any nth term:

$a}_{n}={a}_{1}\times {r}^{n-1}=\frac{7}{8}\times {\left(\frac{3}{5}\right)}^{n-1$

and we can calculate $a}_{5$ just by putting our n=5 in the equation.

$a}_{5}=\frac{7}{8}\times {\left(\frac{3}{5}\right)}^{4}=\frac{567}{500$

$a}_{n}={a}_{1}\times {r}^{n-1$

and for now what we have to do is to figure out what is our first term ( $a}_{1$ ) and we can do it easily cause we have $a}_{4$:

$a}_{4}=\frac{189}{1000}={a}_{1}\times {\left(\frac{3}{5}\right)}^{3$

and we have to solve this equation for $a}_{1$

$a}_{1}=\frac{\frac{189}{1000}}{\frac{{3}^{3}}{{5}^{3}}}=\frac{189\times 125}{1000\times 27}=\frac{7}{8$

Now we can rewrite our equation for any nth term:

$a}_{n}={a}_{1}\times {r}^{n-1}=\frac{7}{8}\times {\left(\frac{3}{5}\right)}^{n-1$

and we can calculate $a}_{5$ just by putting our n=5 in the equation.

$a}_{5}=\frac{7}{8}\times {\left(\frac{3}{5}\right)}^{4}=\frac{567}{500$

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