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Find the sum of the first five terms of the geometric series where [tex]$a_1 = 3$[/tex] and [tex]$r = 2$[/tex].

A. 87
B. 90
C. 93
D. 95

Sagot :

GinnyAnsweridn
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
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