To find the sum of the first five terms of a geometric series where the first term [tex]\( a_1 = 3 \)[/tex] and the common ratio [tex]\( r = 2 \)[/tex], follow these steps:
1. Identify the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series:
[tex]\[ S_n = a_1 \frac{1 - r^n}{1 - r} \][/tex]
where:
- [tex]\( S_n \)[/tex] is the sum of the first [tex]\( n \)[/tex] terms,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the number of terms.
2. Substitute the given values into the formula:
- [tex]\( a_1 = 3 \)[/tex]
- [tex]\( r = 2 \)[/tex]
- [tex]\( n = 5 \)[/tex]
Plugging these values into the formula, we get:
[tex]\[
S_5 = 3 \frac{1 - 2^5}{1 - 2}
\][/tex]
3. Simplify the expression inside the formula:
[tex]\[
2^5 = 32
\][/tex]
[tex]\[
1 - 32 = -31
\][/tex]
[tex]\[
1 - 2 = -1
\][/tex]
4. Compute the sum:
[tex]\[
S_5 = 3 \frac{-31}{-1}
\][/tex]
[tex]\[
S_5 = 3 \times 31
\][/tex]
[tex]\[
S_5 = 93
\][/tex]
Therefore, the sum of the first five terms of the geometric series is [tex]\( 93 \)[/tex].
Among the given choices, the correct answer is:
- 93
