Question

The first term of a geometric sequence is 5
and the common ratio is −3.


What is the sum of the first four terms?

(2 points)

Answer 1

In a geometric sequence, the terms can be calculated using the formula:

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

where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

Given that the first term \( a_1 = 5 \) and the common ratio \( r = -3 \), we can find the first four terms:

1. **First term** \( a_1 = 5 \)
2. **Second term** \( a_2 = a_1 \cdot r = 5 \cdot (-3) = -15 \)
3. **Third term** \( a_3 = a_2 \cdot r = -15 \cdot (-3) = 45 \)
4. **Fourth term** \( a_4 = a_3 \cdot r = 45 \cdot (-3) = -135 \)

Now, we can sum these four terms:

\[
\text{Sum} = a_1 + a_2 + a_3 + a_4 = 5 + (-15) + 45 + (-135)
\]

Calculating it step-by-step:

\[
5 - 15 + 45 - 135 = -10 + 45 - 135 = 35 - 135 = -100
\]

Thus, the sum of the first four terms is:

\[
\boxed{-100}
\]