Question 5 of 10

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.

Answer: ____



Answer :

To find the sum of the first five terms of a geometric series with a first term \( a_1 = 20 \) and a common ratio \( r = \frac{1}{4} \), we can use the formula for the sum of the first \( n \) terms of a geometric series:

[tex]\[ S_n = a_1 \frac{1 - r^n}{1 - r} \][/tex]

Given that \( n = 5 \), \( a_1 = 20 \), and \( r = \frac{1}{4} \), we can substitute these values into the formula to find \( S_5 \):

[tex]\[ S_5 = 20 \cdot \frac{1 - \left(\frac{1}{4}\right)^5}{1 - \frac{1}{4}} \][/tex]

First, we need to calculate \( \left(\frac{1}{4}\right)^5 \):

[tex]\[ \left( \frac{1}{4} \right)^5 = \frac{1}{4 \times 4 \times 4 \times 4 \times 4} = \frac{1}{1024} \][/tex]

So the expression becomes:

[tex]\[ S_5 = 20 \cdot \frac{1 - \frac{1}{1024}}{1 - \frac{1}{4}} \][/tex]

Next, we simplify \( 1 - \frac{1}{1024} \):

[tex]\[ 1 - \frac{1}{1024} = \frac{1024}{1024} - \frac{1}{1024} = \frac{1023}{1024} \][/tex]

Now, we simplify the denominator \( 1 - \frac{1}{4} \):

[tex]\[ 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \][/tex]

Now the sum \( S_5 \) is:

[tex]\[ S_5 = 20 \cdot \frac{\frac{1023}{1024}}{\frac{3}{4}} \][/tex]

To divide by a fraction, we multiply by its reciprocal:

[tex]\[ S_5 = 20 \cdot \frac{1023}{1024} \times \frac{4}{3} = 20 \cdot \frac{1023 \times 4}{1024 \times 3} \][/tex]

Simplifying the fractions:

[tex]\[ 1023 \times 4 = 4092 \][/tex]
[tex]\[ 1024 \times 3 = 3072 \][/tex]

So:

[tex]\[ S_5 = 20 \cdot \frac{4092}{3072} \][/tex]

Next, we need to simplify \( \frac{4092}{3072} \):

[tex]\[ \frac{4092}{3072} = \frac{4092 \div 4}{3072 \div 4} = \frac{1023}{768} \][/tex]

Thus, the expression is:

[tex]\[ S_5 = 20 \cdot \frac{1023}{768} \][/tex]

This is equal to:

[tex]\[ S_5 = 20 \times \frac{2046}{1536} = \frac{2046}{1536} \times 20 = \frac{49152}{1536} = 32 \][/tex]

Expressing this result back in proper terms:

The sum of the first five terms of the series is 26.640625. However, as needed we convert it to improper fraction result by recalculating :
Sum is: \( \frac{2046}{1536} times 20 = 26.640625\ \ )
Answer = 853/32