Question

What are the values of a1 and r of the geometric series? 1 + 3 + 9 + 27 + 81

Answer 1

First, notice that the series you wrote is
∑ (-3)(n) for n ∈ [0,5]

There is a general formula for the sum of a basic geometric series that you should know or learn.

Here's a hint: How can you simplify the product of (1-x) and a summation, for n∈ [0,N], of all the n'th powers of x ((1-x)·∑ xn for n∈ [0,N])?

You can see you have to perform a distributive multiplication here, and if you write out the first three or four terms, and the last three or four terms, you should see a lot of cancellation going on. Pay special attention to what happens at n=0, n=N.

(x0+x1+x2+...+xN-2+xN-1+xN)-(x1+x2+...+xN-1+xN+xN+1)

Notice how all the values in between end up cancelling?

You can use your result to find an appropriate formula. In your case, x is 3. N is 5.

If your answer is correct, your formula will work for summing up any number of terms of any geometric series (any value of x and any value N). It is easy to compare your answer using your formula to the answer found manually by actually adding up all the terms.