What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9?
well, assuming you want |r|<1, we have
S = 9/(1-r)
r S
0.1 10
0 9
-1/8 8
-2/7 7
-1/2 6
-4/5 5
To find the smallest integer that can possibly be the sum of an infinite geometric series with a first term of 9, we need to determine the common ratio of the series.
The formula for the sum of an infinite geometric series is given by:
S = a / (1 - r)
where:
S = sum of the infinite series
a = first term of the series
r = common ratio of the series
In this case, we have the first term a = 9. The common ratio r can be any real number between -1 and 1, exclusive, for the series to converge.
To find the smallest possible sum, we need to find the smallest value of r that gives an integer result for the sum S.
Let's try different values of r within the range (-1, 1):
For r = 1/2: S = 9 / (1 - 1/2) = 9 / (1/2) = 18, which is an integer.
For r = 1/3: S = 9 / (1 - 1/3) = 9 / (2/3) = 27/2, which is not an integer.
For r = 1/4: S = 9 / (1 - 1/4) = 9 / (3/4) = 36/3 = 12, which is an integer.
For r = 1/5: S = 9 / (1 - 1/5) = 9 / (4/5) = 45/4, which is not an integer.
By testing different values for r, we find that the smallest value of r that gives an integer result for the sum S is 1/2. Therefore, the smallest integer that can possibly be the sum of the series is 18.