Given a geometric series with the first term [tex]\(a_T = 10\)[/tex] and common ratio [tex]\(r = \frac{1}{5}\)[/tex], we need to find the sum of the first five terms, expressed as an improper fraction in lowest terms.
To find the sum of the first [tex]\(n\)[/tex] terms of a geometric series, we use the formula:
[tex]\[ S_n = a_T \frac{1 - r^n}{1 - r} \][/tex]
For our specific problem:
- The first term [tex]\(a_T = 10\)[/tex]
- The common ratio [tex]\(r = \frac{1}{5}\)[/tex]
- The number of terms [tex]\(n = 5\)[/tex]
First, calculate [tex]\(r^5\)[/tex]:
[tex]\[ r^5 = \left(\frac{1}{5}\right)^5 = \frac{1}{3125} \][/tex]
Next, compute [tex]\(1 - r^5\)[/tex]:
[tex]\[ 1 - r^5 = 1 - \frac{1}{3125} = \frac{3125}{3125} - \frac{1}{3125} = \frac{3124}{3125} \][/tex]
Now, use the sum formula:
[tex]\[ S_5 = 10 \cdot \frac{\frac{3124}{3125}}{1 - \frac{1}{5}} \][/tex]
Simplify the denominator [tex]\(1 - \frac{1}{5}\)[/tex]:
[tex]\[ 1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5} \][/tex]
Substitute back into the sum formula:
[tex]\[ S_5 = 10 \cdot \frac{\frac{3124}{3125}}{\frac{4}{5}} \][/tex]
Simplify the fraction inside the sum formula:
[tex]\[ S_5 = 10 \cdot \left( \frac{3124}{3125} \cdot \frac{5}{4} \right) \][/tex]
Multiply the fractions:
[tex]\[ S_5 = 10 \cdot \frac{3124 \cdot 5}{3125 \cdot 4} = 10 \cdot \frac{15620}{12500} = 10 \cdot \frac{1562}{1250} \][/tex]
Now, simplify to lowest terms:
[tex]\[ \frac{1562}{1250} = \frac{781}{625} \][/tex]
Thus, the sum of the first five terms of the geometric series is:
[tex]\[ S_5 = 10 \cdot \frac{781}{625} = \frac{7810}{625} \][/tex]
Finally, express the improper fraction in lowest terms:
[tex]\[ \frac{7810}{625} \][/tex]
Therefore, the sum of the first five terms of the geometric series is:
[tex]\[ \frac{7810}{625} \][/tex]
Answer here:
[tex]\[ \frac{7810}{625} \][/tex]
