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What is the sum of the first five terms of a geometric series with [tex]$a_1=10$[/tex] and [tex]$r=\frac{1}{5}$[/tex]?

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

Sagot :

GinnyAnswer
Sure! Let's break this down step by step.

Given:
- First term, [tex]\( a_1 = 10 \)[/tex]
- Common ratio, [tex]\( r = \frac{1}{5} \)[/tex]
- Number of terms, [tex]\( n = 5 \)[/tex]

To find the sum of the first five terms of a geometric series, we 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]

Plugging in the given values:

1. Compute [tex]\( r^n \)[/tex]:

[tex]\[ r^5 = \left(\frac{1}{5}\right)^5 = \frac{1}{5^5} = \frac{1}{3125} \][/tex]

2. Calculate the term [tex]\( 1 - r^n \)[/tex]:

[tex]\[ 1 - r^5 = 1 - \frac{1}{3125} = \frac{3125}{3125} - \frac{1}{3125} = \frac{3124}{3125} \][/tex]

3. Substitute into the [tex]\( S_n \)[/tex] formula:

[tex]\[ S_5 = 10 \times \frac{\frac{3124}{3125}}{\frac{4}{5}} \][/tex]

4. Simplify the division:

[tex]\[ \frac{\frac{3124}{3125}}{\frac{4}{5}} = \frac{3124}{3125} \times \frac{5}{4} = \frac{3124 \times 5}{3125 \times 4} = \frac{15620}{12500} \][/tex]

5. Simplify the fraction:

[tex]\[ S_5 = \frac{15620}{12500} \][/tex]

To simplify [tex]\( \frac{15620}{12500} \)[/tex] to its lowest terms:

[tex]\[ \frac{15620 \div 20}{12500 \div 20} = \frac{781}{625} \][/tex]

Thus, the sum of the first five terms of the geometric series is:

[tex]\[ S_5 = \frac{781}{625} \][/tex]

So, [tex]\( \frac{781}{625} \)[/tex] is the sum of the first five terms of the geometric series.
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