Sure! Let's break this down step by step.
Given:
- First term, [tex]\( a_1 = 10 \)[/tex]
- Common ratio, [tex]\( r = \frac{1}{5} \)[/tex]
- Number of terms, [tex]\( n = 5 \)[/tex]
To find the sum of the first five terms of a geometric series, we use 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]
Plugging in the given values:
1. Compute [tex]\( r^n \)[/tex]:
[tex]\[
r^5 = \left(\frac{1}{5}\right)^5 = \frac{1}{5^5} = \frac{1}{3125}
\][/tex]
2. Calculate the term [tex]\( 1 - r^n \)[/tex]:
[tex]\[
1 - r^5 = 1 - \frac{1}{3125} = \frac{3125}{3125} - \frac{1}{3125} = \frac{3124}{3125}
\][/tex]
3. Substitute into the [tex]\( S_n \)[/tex] formula:
[tex]\[
S_5 = 10 \times \frac{\frac{3124}{3125}}{\frac{4}{5}}
\][/tex]
4. Simplify the division:
[tex]\[
\frac{\frac{3124}{3125}}{\frac{4}{5}} = \frac{3124}{3125} \times \frac{5}{4} = \frac{3124 \times 5}{3125 \times 4} = \frac{15620}{12500}
\][/tex]
5. Simplify the fraction:
[tex]\[
S_5 = \frac{15620}{12500}
\][/tex]
To simplify [tex]\( \frac{15620}{12500} \)[/tex] to its lowest terms:
[tex]\[
\frac{15620 \div 20}{12500 \div 20} = \frac{781}{625}
\][/tex]
Thus, the sum of the first five terms of the geometric series is:
[tex]\[
S_5 = \frac{781}{625}
\][/tex]
So, [tex]\( \frac{781}{625} \)[/tex] is the sum of the first five terms of the geometric series.
