How would I find the sum of the geometric series given?
1/5 - 1/25 + 1/125
1/5 - 1/25 + 1/125 =
LCD = 125
25/125 - 5/125 + 1/125 = 21/125.
To find the sum of a geometric series, you need to use the formula for the sum of an infinite geometric series. The formula is:
S = a / (1 - r)
Where:
- S represents the sum of the series
- a represents the first term of the series
- r represents the common ratio of the series
In this case, using the formula:
a = 1/5 (the first term)
r = -1/5 (the common ratio)
Now substitute these values into the formula:
S = (1/5) / (1 - (-1/5))
Simplify the denominator:
S = (1/5) / (1 + 1/5)
Find the common denominator:
S = (1/5) / (5/5 + 1/5)
Simplify:
S = (1/5) / (6/5)
Divide the fraction:
S = (1/5) x (5/6)
Simplify:
S = 1/6
Therefore, the sum of the given geometric series is 1/6.
To find the sum of a geometric series, you need to use the formula:
S = a / (1 - r)
Where:
- S represents the sum of the series
- a represents the first term of the series
- r represents the common ratio between consecutive terms
In your given series, the first term is 1/5 and the common ratio is -1/5 (because each term is multiplied by -1/5 to get the next term). Let's substitute these values into the formula:
S = (1/5) / (1 - (-1/5))
To simplify it further:
S = (1/5) / (1 + 1/5)
Now, let's find a common denominator:
S = (1/5) / (5/5 + 1/5)
S = (1/5) / (6/5)
Next, to divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
S = (1/5) * (5/6)
S = 5/30
Finally, we simplify the fraction:
S = 1/6
Therefore, the sum of the given geometric series is 1/6.