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.

S = 9/(1-r)

If r>=0, you can see that 9/1=9 would be the smallest possible sum, if r=0. That is, if the sequence is
9 0 0 0 0 0 ......

Any other value for r will produce nonzero terms, and therefore a larger sum.

You can investigate what happens when r < 0.

To find the smallest integer that can possibly be the sum of an infinite geometric series, we need to understand the formula for the sum of an infinite geometric series.

The formula for the sum of an infinite geometric series is given by:

S = a / (1 - r)

Where:
S = the sum of the series
a = the first term of the series
r = the common ratio

In this case, we are given that the first term (a) is 9. Therefore, we can rewrite the formula as:

S = 9 / (1 - r)

Now, we need to find the smallest value for S that results in an integer. To achieve this, we want the denominator (1 - r) to be a factor of the numerator (9).

Since we are looking for the smallest integer, we can start by setting r = 1 since it yields the smallest denominator.

Plugging r = 1 into the formula, we get:

S = 9 / (1 - 1)
S = 9 / 0

However, dividing by zero is undefined, so r = 1 does not give us a valid sum.

Next, we need to consider another value for r. Let's set r = 2, which is commonly used in geometric series.

Plugging r = 2 into the formula, we get:

S = 9 / (1 - 2)
S = 9 / (-1)
S = -9

Here, we get a negative integer. We want the smallest positive integer, so we need to continue exploring other values of r.

Let's set r = 1/2.

Plugging r = 1/2 into the formula, we get:

S = 9 / (1 - 1/2)
S = 9 / (1/2)
S = 9 * (2/1)
S = 18

Finally, we have obtained a positive integer. Therefore, the smallest integer that can possibly be the sum of an infinite geometric series with a first term of 9 is 18.

To find the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9, we need to understand the formula for the sum of an infinite geometric series.

The formula for the sum of an infinite geometric series is given by:

S = a / (1 - r)

where:
S is the sum
a is the first term
r is the common ratio

In this case, the first term (a) is 9.

To find the smallest integer value for S, we need to look for the smallest value of r that results in an integer value for S.

Let's consider different values for r and calculate the corresponding values of S until we find an integer:

1. r = 1/2:
S = 9 / (1 - 1/2) = 18

2. r = 1/3:
S = 9 / (1 - 1/3) = 13.5

3. r = 1/4:
S = 9 / (1 - 1/4) = 12

4. r = 1/5:
S = 9 / (1 - 1/5) = 11.25

5. r = 1/6:
S = 9 / (1 - 1/6) = 10.2857...

We can continue the process for different values of r, but we notice that there is a pattern. As the value of r approaches 1, the value of S increases.

If r = 1, the denominator (1 - r) will be zero, resulting in division by zero, and the sum S cannot be calculated.

Therefore, the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9 is 12.