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What is the sum of the first five terms of a geometric series with [tex]a_1=20[/tex] and [tex]r=\frac{1}{4}[/tex]?

Express your answer as an improper fraction in lowest terms without using spaces.


Sagot :

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]