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
[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
