SOLUTION: 3) Use the geometric sequence of numbers 1, 1/2, 1/4, 1/8,…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 to get from term to term. So
r=nth term/(n-1) term


The sequence is cut in half each term, so the sequence is

b)
The sum of a geometric series is
where a=1
So plug in n=10 to find the sum of the first 10 partial sums


So the sum of the first ten terms is or 1.99805 approximately
c)
Use the same formula to find the sum of the 1st 12 terms
where a=1
So plug in n=12 to find the sum of the first 12 partial sums


So the sum of the first twelve terms is or 1.99951 approximately

d)
It appears that the sums are approaching a finite number of 2. This is because each term is getting smaller and smaller. This observation is justified by the fact that if then the infinite series will approach a finite number. In other words
If (the magnitude of r has to be less than 1) then,
Where S is the infinite series. So if we let a=1 and r=1/2 we get


So this verifies that our series approaches 2. Hope this helps.