Question

Which of the following correctly uses the formula for the finite geometric series to derive the sum of the first seven terms of the geometric series 14 + 21 + 31.5 + 47.25 + . . . ?(1 point)
Responses

S7=14−14(1.5)61−1.5
upper S subscript 7 baseline equals Start Fraction 14 minus 14 left parenthesis 1.5 right parenthesis superscript 6 baseline over 1 minus 1.5 End Fraction

S7=14−14(7)71−7
upper S subscript 7 baseline equals Start Fraction 14 minus 14 left parenthesis 7 right parenthesis superscript 7 baseline over 1 minus 7 End Fraction

S7=14−14(1.5)71+1.5
upper S subscript 7 baseline equals Start Fraction 14 minus 14 left parenthesis 1.5 right parenthesis superscript 7 baseline over 1 plus 1.5 End Fraction

S7=14−14(1.5)71−1.5

Answer 1

To find the sum of the first seven terms of the geometric series given as 14, 21, 31.5, 47.25, ..., we first need to identify the first term \( a \) and the common ratio \( r \).

1. **Identify the first term**:
The first term \( a \) is 14.

2. **Identify the common ratio \( r \)**:
We can find the common ratio by dividing the second term by the first term:
\[
r = \frac{21}{14} = 1.5
\]

3. **Use the formula for the sum of the first \( n \) terms of a geometric series**:
The formula for the sum of the first \( n \) terms of a geometric series is:
\[
S_n = a \frac{1 - r^n}{1 - r}
\]
where \( S_n \) is the sum of \( n \) terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

4. **Substituting values**:
We need to find \( S_7 \):
\[
S_7 = 14 \frac{1 - (1.5)^7}{1 - 1.5}
\]

Now, let's evaluate the final expression:

- The denominator \( 1 - 1.5 = -0.5 \).
- Thus,
\[
S_7 = 14 \frac{1 - (1.5)^7}{-0.5}
\]
which simplifies to:
\[
S_7 = -28(1 - (1.5)^7)
\]

Now let's compare this with the given options. The correct option must have the form \( S_7 = \frac{14(1 - (1.5)^7)}{1 - 1.5} \).

Among the provided options, the correct one is:

\[
S_7 = \frac{14 - 14(1.5)^7}{1 - 1.5}
\]

So the correct answer is:
\[
S_7 = \frac{14 - 14(1.5)^7}{1 - 1.5}
\]

This matches with the last option you provided:

- **S7=14 − 14(1.5)7 / 1 − 1.5**

Thus, the correct choice is:

**S7=14−14(1.5)71−1.5**