SOLUTION: 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: r=1/2 divided by 1, which

Question 84140: 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: r=1/2 divided by 1,
which is 1/2(fraction)
Is this correct
b) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms? Round your answer to 4 decimals.
show work
1st term is 1X1/2=1/2
2nd term is 1/2X1/2=1/4
3rd term is 1/4X1/2=1/8
4th term is 1/8X1/2=1/16
5th term is 1/16X1/2=1/32
6th term is 1/32X1/2=1/64
7th term is 1/64X1/2=1/128
8th term is 1/128X1/2=1/256
9th term is 1/256X1/2=1/512
10th term is 1/512X1/2=1,024
c)Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 12 terms? Round to 4 decimals.
show work
11th term is 1/1,024X1/2=2,048
12 term is 1/2,048X1/2=4,096
d) What observation can you make about these sums? In particular, what whole number does it appear that the sum will always be smaller than?
1
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)
The ratio r is the factor to get from term to term. So
r=nth term/(n-1) term

So the ratio is (you are correct)
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.

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