SOLUTION: 2) Use the geometric sequence of numbers 1, 3, 9, 27, … to find the following: a) What is r, the ratio between 2 consecutive terms? Answer: Show work in this space

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a)
The ratio r is the factor needed to go from term to term. To find the factor, divide any term by its previous term. So I chose 3 as the first term to be divided by 1
Ratio r: pick any nth term and any previous term, such as the 2nd and 1st term.
I can also do it with 27 and 9 and it will still give me the same value

So r=3


b) Since we are multiplying by r each term, our sequence is simply



So to find the nth term (), we simply multiply the previous term () by r.

So if we start at 1, to get to 3 we multiply 1 by r=3



Now to go from 3 to 9 we multiply by r again



Now to go from 9 to 27 we multiply by r again



Notice that for the 1st term 3 we have only 1 r, 2nd term we have 2 r's, etc. So the term we have determines the number of r's. In other words, the nth term is



So our sequence is

Now let n=9 to find the 10th term (we started at n=0)



So the 10th term is 19,683


c) The sum of the first ten terms can be found by using

So let r=3



So the sum of the first ten terms are 29,524