The second term of a geometric sequence is 12 . The fourth term is 108. What is the fifth term?

Question

UpStudy AI · Accepted Answer

Let the first term be $$a$$ and the common ratio be $$r$$. Then the terms of the geometric sequence are:
- First term: $$a$$
- Second term: $$ar$$
- Third term: $$ar^2$$
- Fourth term: $$ar^3$$
- Fifth term: $$ar^4$$

We are given:
$$
ar = 12 \quad \text{and} \quad ar^3 = 108
$$

Step 1. Divide the fourth term by the second term to eliminate $$a$$:
$$
\frac{ar^3}{ar} = \frac{108}{12}
$$
$$
r^2 = 9
$$
$$
r = 3 \quad \text{or} \quad r = -3
$$

Step 2. Find $$a$$ using $$ar = 12$$:

For $$r = 3$$:
$$
a = \frac{12}{3} = 4
$$

For $$r = -3$$:
$$
a = \frac{12}{-3} = -4
$$

Step 3. Calculate the fifth term $$ar^4$$:

For $$r = 3$$ and $$a = 4$$:
$$
ar^4 = 4 \times 3^4 = 4 \times 81 = 324
$$

For $$r = -3$$ and $$a = -4$$:
$$
ar^4 = -4 \times (-3)^4 = -4 \times 81 = -324
$$

Since both options satisfy the given conditions, there are two possible fifth terms: $$324$$ or $$-324$$.

In many contexts, the positive ratio is assumed, so the fifth term is:
$$
\boxed{324}
$$
The fifth term is 324.

The second term of a geometric sequence is 12 . The fourth term is 108. What is the fifth term?

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Algebra Questions