A geometric sequence has a second term of -\frac{3}{2} and a fifth term of 12. What is the… — ACT

A geometric sequence has a second term of $-\frac{3}{2}$ and a fifth term of $12$ . What is the eighth term of this sequence?

-96

The correct term is

-96

. First, find the common ratio

r

by taking the ratio of the fifth term to the second term:

\frac{a_5}{a_2} = \frac{a_1 r^4}{a_1 r} = r^3

. Substituting the given values,

r^3 = \frac{12}{-3/2} = -8

, which gives

r = -2

. To find the eighth term, multiply the fifth term by the common ratio cubed:

a_8 = a_5 r^3 = 12 \times (-2)^3 = 12 \times (-8) = -96

Set up the ratio between the fifth term and the second term using the geometric sequence formula

a_n = a_1 r^{n-1}

\frac{a_5}{a_2} = \frac{a_1 r^4}{a_1 r} = r^3

This allows us to isolate the common ratio

r

without needing to calculate the first term

a_1

first.

Substitute the given values into the ratio and solve for

r

r^3 = \frac{12}{-3/2} = 12 \times \left(-\frac{2}{3}\right) = -8 \Rightarrow r = -2

Finding the common ratio is necessary to compute any subsequent terms in the sequence.

Use the common ratio to find the eighth term

a_8

by multiplying the fifth term

a_5

r^3

a_8 = a_5 r^{8-5} = 12 \times (-2)^3 = 12 \times (-8) = -96

Since

a_8 = a_5 r^3

, multiplying

12

-8

directly gives the eighth term.

Finding terms in a geometric sequence using the common ratio.

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