SOLUTION: What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9?

Question 1094743: What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

That is an interesting question. I have worked hundreds of problems involving infinite geometric sequences and series; but this is a new twist to me -- and so it caught my interest.

The sum of an infinite geometric series with first term a and common ratio r (|r|<1) is

We want to know the smallest integer the sum can be if the first term is 9. A formal algebraic method for finding the answer is probably possible; but it seems a "brute force" method might get to the answer faster and with less work.

So start with a sum of 1 and see if you get a converging infinite geometric sequence; if that doesn't work, try 2; and so on.

S = 1:

|r|>1; won't work

S = 2:

still won't work

I won't show the details of the calculations for the next few integer values of the sum. By looking at the algebra done in solving the equation for the different values of S, you might see the pattern in the values of r we get for the different integer values of the sum. They are
S = 3 --> r = -6/3 = -2
S = 4 --> r = -5/4
S = 5 --> r = -4/5

5 is the smallest integer sum which gives us a value of r that satisfies |r|<1, so that is our answer: the smallest integer sum for an infinite geometric series with first term 9 is 5.

The sequence itself is
9, -36/5, 144/25, -576/125, ...

Checking, the infinite sum is

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