To find the sum of the first five terms of a geometric series with a first term [tex]\(a_1 = 20\)[/tex] and a common ratio [tex]\(r = \frac{1}{4}\)[/tex], we can 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]
where:
- [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms,
- [tex]\(a_1\)[/tex] is the first term, which is 20,
- [tex]\(r\)[/tex] is the common ratio, which is [tex]\(\frac{1}{4}\)[/tex],
- [tex]\(n\)[/tex] is the number of terms, which is 5.
First, we need to find [tex]\(r^n\)[/tex]:
[tex]\[ r^n = \left(\frac{1}{4}\right)^5 = \frac{1}{1024} \][/tex]
Next, we substitute [tex]\(r\)[/tex], [tex]\(a_1\)[/tex], and [tex]\(r^n\)[/tex] back into the sum formula:
[tex]\[ S_5 = 20 \frac{1 - \frac{1}{1024}}{1 - \frac{1}{4}} \][/tex]
Simplify the denominator:
[tex]\[ 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \][/tex]
Now, simplify the numerator:
[tex]\[ 1 - \frac{1}{1024} = \frac{1024}{1024} - \frac{1}{1024} = \frac{1023}{1024} \][/tex]
Putting it all together:
[tex]\[ S_5 = 20 \frac{\frac{1023}{1024}}{\frac{3}{4}} \][/tex]
We need to divide the fractions:
[tex]\[ \frac{\frac{1023}{1024}}{\frac{3}{4}} = \frac{1023}{1024} \times \frac{4}{3} = \frac{1023 \times 4}{1024 \times 3} = \frac{4092}{3072} \][/tex]
Reduce this fraction:
[tex]\[ \frac{4092 \div 4}{3072 \div 4} = \frac{1023}{768} \][/tex]
Finally, multiply by 20:
[tex]\[ S_5 = 20 \times \frac{1023}{768} = \frac{20 \times 1023}{768} = \frac{20460}{768} \][/tex]
Simplify:
[tex]\[ \frac{20460 \div 4}{768 \div 4} = \frac{5115}{192} \][/tex]
Therefore, the sum of the first five terms of the series is:
[tex]\[ \boxed{\frac{5115}{192}} \][/tex]
