What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9? Please explain in steps.
To find the smallest integer that can possibly be the sum of an infinite geometric series, we need to analyze the formula for the sum of an infinite geometric series.
The formula for the sum of an infinite geometric series 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 between consecutive terms
In this case, the first term (a) is given as 9.
Now, we need to determine the value of the common ratio (r) that would yield the smallest integer sum. For an integer sum to exist, the common ratio must be a fraction in the range (0, 1).
To minimize the sum, we can set the common ratio (r) to the smallest fractional value within that range, which is 1/10 (equivalent to 0.1).
Substituting the values into the formula:
S = 9 / (1 - 1/10)
S = 9 / (9/10)
S = 9 * (10/9)
S = 10
Hence, the smallest integer that can be the sum of the infinite geometric series with a first term of 9 is 10.