Certainly! Let's calculate the sum of the geometric series
[tex]\[
\sum_{k=1}^4 6^{k-1}.
\][/tex]
To find the sum of the first [tex]\( n \)[/tex] terms of a geometric series, we use the formula:
[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, and
- [tex]\( n \)[/tex] is the number of terms.
For the series [tex]\(\sum_{k=1}^4 6^{k-1}\)[/tex], we can identify the components as follows:
- The first term [tex]\( a \)[/tex] is [tex]\( 6^{1-1} = 6^0 = 1 \)[/tex],
- The common ratio [tex]\( r \)[/tex] is 6,
- The number of terms [tex]\( n \)[/tex] is 4.
Using the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series, we can substitute the values:
[tex]\[
S_4 = 1 \cdot \frac{6^4 - 1}{6 - 1}.
\][/tex]
Now, calculate [tex]\( 6^4 \)[/tex]:
[tex]\[
6^4 = 1296.
\][/tex]
Next, substitute this back into the formula:
[tex]\[
S_4 = 1 \cdot \frac{1296 - 1}{5} = \frac{1295}{5}.
\][/tex]
Now, perform the division:
[tex]\[
\frac{1295}{5} = 259.0.
\][/tex]
Therefore, the sum of the geometric series [tex]\(\sum_{k=1}^4 6^{k-1}\)[/tex] is:
[tex]\[
1 + 6 + 36 + 216 = 259.0.
\][/tex]
So, the sum is [tex]\( 259.0 \)[/tex].
