Answer :
To find the sum of a geometric series, we can use the formula for the sum of the first [tex]\(n\)[/tex] terms, given by:
[tex]\[ S_n = a \frac{r^n - 1}{r - 1} \][/tex]
where:
- [tex]\( a \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the number of terms.
Given:
- [tex]\( a = 1 \)[/tex]
- [tex]\( r = 3 \)[/tex]
- [tex]\( n = 8 \)[/tex]
Let's plug these values into the formula and calculate each step:
1. Calculate [tex]\( r^n \)[/tex]:
[tex]\[ r^n = 3^8 \][/tex]
2. Compute [tex]\( 3^8 \)[/tex]:
[tex]\[ 3^8 = 6561 \][/tex]
3. Subtract 1 from [tex]\( 3^8 \)[/tex]:
[tex]\[ 3^8 - 1 = 6561 - 1 = 6560 \][/tex]
4. Divide by [tex]\( r - 1 \)[/tex]:
[tex]\[ \frac{6560}{3 - 1} = \frac{6560}{2} = 3280 \][/tex]
Therefore, the sum of the geometric series is:
[tex]\[ S_8 = 3280 \][/tex]
Thus, the correct answer is:
[tex]\[ 3280 \][/tex]
[tex]\[ S_n = a \frac{r^n - 1}{r - 1} \][/tex]
where:
- [tex]\( a \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the number of terms.
Given:
- [tex]\( a = 1 \)[/tex]
- [tex]\( r = 3 \)[/tex]
- [tex]\( n = 8 \)[/tex]
Let's plug these values into the formula and calculate each step:
1. Calculate [tex]\( r^n \)[/tex]:
[tex]\[ r^n = 3^8 \][/tex]
2. Compute [tex]\( 3^8 \)[/tex]:
[tex]\[ 3^8 = 6561 \][/tex]
3. Subtract 1 from [tex]\( 3^8 \)[/tex]:
[tex]\[ 3^8 - 1 = 6561 - 1 = 6560 \][/tex]
4. Divide by [tex]\( r - 1 \)[/tex]:
[tex]\[ \frac{6560}{3 - 1} = \frac{6560}{2} = 3280 \][/tex]
Therefore, the sum of the geometric series is:
[tex]\[ S_8 = 3280 \][/tex]
Thus, the correct answer is:
[tex]\[ 3280 \][/tex]
