Question

What is the fifth term of the geometric sequence?

5,15,45,...

(1 point)
Responses

1215
1215

405
405

3645
3645

1875
1875
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Answers

Answered by GPT-4o mini
To find the fifth term of the geometric sequence given (5, 15, 45,...), we first need to identify the common ratio.

The common ratio \( r \) can be calculated by dividing the second term by the first term:

\[
r = \frac{15}{5} = 3
\]

We can also confirm this by dividing the third term by the second term:

\[
r = \frac{45}{15} = 3
\]

Now that we know the common ratio \( r = 3 \), we can apply it to find the subsequent terms in the sequence.

The formula for the \( n \)-th term of a geometric sequence is:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Where:
- \( a_1 = 5 \) (the first term)
- \( r = 3 \) (the common ratio)

We want to find the fifth term (\( n = 5 \)):

\[
a_5 = 5 \cdot 3^{(5-1)} = 5 \cdot 3^4
\]

Calculating \( 3^4 \):

\[
3^4 = 81
\]

Now substituting back into the equation:

\[
a_5 = 5 \cdot 81 = 405
\]

Therefore, the fifth term of the geometric sequence is \(\boxed{405}\).
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