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

To find the fifth term of the geometric sequence with the given initial terms of 5, 15, and 45, we start by understanding the nature of a geometric sequence. A geometric sequence is defined by the property that each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

In this geometric sequence, we can identify the common ratio by dividing the second term by the first term:

Common Ratio (r) = Second Term (15) / First Term (5) = 3

We can verify this by checking the ratio for the third term:

Third Term (45) / Second Term (15) = 3

This confirms that the common ratio (r) is indeed 3.

Using the common ratio, we can find subsequent terms in the sequence:

• First Term (a₁) = 5
• Second Term (a₂) = a₁ * r = 5 * 3 = 15
• Third Term (a₃) = a₂ * r = 15 * 3 = 45
• Fourth Term (a₄) = a₃ * r = 45 * 3 = 135

Now, to find the fifth term (a₅):

a5 = a4 * r = 135 * 3 = 405

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