Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are 36 and 2304, respectively.

Answer 1

Answer:

$a_n=9(4^{n-1})$

Step-by-step explanation:

we know that

In a Geometric Sequence each term is found by multiplying the previous term by a constant, called the common ratio (r)

In this problem we have

$a_2=36\\ a_5=2,304$

Remember that

$a_2=a_1(r)$ -----> $36=a_1(r)$ -----> equation A

$a_5=a_4(r)$

$a_5=a_3(r^{2})$

$a_5=a_2(r^{3})$

Substitute the values of a_5 and a_2 and solve for r

$2,304=36(r^{3})$

$r^{3}=2,304/36$

$r^{3}=64$

$r=4$

Find the value of a_1 in equation A

$36=a_1(4)$

$a_1=9$

therefore

The explicit rule for the nth term is

$a_n=a_1(r^{n-1})$

substitute

$a_n=9(4^{n-1})$