What is the fifth term of the geometric sequence 5, 15, 45?

To find the fifth term of the geometric sequence where the first three terms are 5, 15, and 45, we first need to identify the common ratio of the sequence.

A geometric sequence is defined such that each term after the first is found by multiplying the previous term by a constant known as the common ratio. To find this ratio, we can divide the second term by the first term:

• Common ratio (r) = 15 / 5 = 3

We can check this ratio by dividing the third term by the second term:

• Common ratio (r) = 45 / 15 = 3

Since the ratio is consistent, we confirm that the common ratio (r) is indeed 3.

Next, we can use the formula for the nth term of a geometric sequence, which is:

a_n = a₁ * r^n-1

Here, a₁ is the first term, r is the common ratio, and n is the term number we want to find.

For the fifth term (n = 5), we plug in the values:

• a₅ = 5 * 3^5-1 = 5 * 3⁴
• a₅ = 5 * 81 = 405

Thus, the fifth term of the geometric sequence is 405.