Find the indicated $n$th term of the geometric sequence.
6th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 8 } = \frac { 4 } { 2187 }$
A) $\frac { 4 } { 729 }$
B) $\frac { 3 } { 1024 }$
C) $\frac { 4 } { 2187 }$
D) $\frac { 4 } { 243 }$
E) $\frac { 4 } { 81 }$

Question

Quizplus · Accepted Answer

We can use the formula for the nth term of a geometric sequence to solve for the common ratio, and then use that to find the 6th term.

The formula for the nth term of a geometric sequence is:

a_n = a_1 * r^(n-1)

where a_1 is the first term and r is the common ratio.

We are given two terms, a_5 and a_8:

a_5 = 4/81
a_8 = 4/2187

We can use these to find the common ratio:

a_8/a_5 = (4/2187)/(4/81) = (4/2187)* (81/4) = (3/7)^3

So the common ratio is (3/7)^3.

Now we can use the formula to find the 6th term:

a_6 = a_1 * r^(6-1) = a_1 * r^5

We know that a_5 = a_1 * r^4, so we can use that to solve for a_1:

a_1 = a_5 / r^4 = (4/81) / ((3/7)^3)^4 = (4/81) / (3^12/7^9) = (4/81) * (7^9/3^12)

Now we can use this value of a_1 to find a_6:

a_6 = a_1 * r^5 = ((4/81) * (7^9/3^12)) * ((3/7)^3)^5 = (4/81) * (7^6/3^9) = 4/243

Therefore, the 6th term is 4/243, so the answer is D.

Find the Indicated nnn Th Term of the Geometric Sequence a5=481,a8=42187a _ { 5 } = \frac { 4 } { 81 } , a _ { 8 } = \frac { 4 } { 2187 }a5​=814​,a8​=21874​

Find the Indicated $n$ Th Term of the Geometric Sequence $a _ { 5 } = \frac { 4 } { 81 } , a _ { 8 } = \frac { 4 } { 2187 }$