Use the geometric sequence of numbers
1, 1/2, 1/4, 1/8,...to
find the following:
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a) What is r, the ratio between 2 consecutive terms?
Just divide each given term after the first by the
preceding one and see if you get the same number. If
you do, then you call that number "the common ratio, r".
For 1, 1/2, 1/4, 1/8,... we divide the second term, 1/2,
by the first term 1, like this: 1/2÷1 = 1/2. Then we
divide the third term 1/4, by the second term 1/2, like
this: (1/4)÷(1/2) = (1/4)×(2/1) = 2/4 = 1/2.
Then we divide the fourth term, 1/8, by the third term,
1/4, like this" (1/8)÷(1/4) = (1/8)×(4/1) = 4/8 = 1/2.
Every time we got 1/2. So that means this is a geometric
sequence and the common ratio, r, is 1/2. So r = 1/2.
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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? Please round your answer to
4 decimals.
The formula for the sum, called Sn, of the
first n terms of a geometric sequence is either
of these two equivalent formulas:
Sn = a1(rn - 1)/(r - 1)
or
Sn = a1(1 - rn)/(1 - r)
where a1 stands for the first term, r stands
for the common ratio, and n stands for the number of
term that you want to find. It doesn't matter which of
those formulas you use, because you'll get the same
answer using either one. Normally we use the first one
if |r| > 1 and the second one if |r| < 1, but there is
no rule. I'll use the second one.
Here a1 = 1, r = 1/2, and n = 10 so we plug those in:
Sn = a1(1 - rn)/(1 - r)
S10 = (1)[1 - (1/2)10]/(1 - 1/2)
S10 = (1)(1 - 1/210)/(1/2)
S10 = (1 - 1/210)/(1/2)
S10 = (1 - 1/210)×(2/1)
S10 = 2(1 - 1/210)
S10 = 2 - 2/210
S10 = 2 - 1/29
S10 = 2 - 1/512 = 1.9980
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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? Please round your answer to
4 decimals.
Same as above using 12 for n instead of 10
a1 = 1, r = 1/2, and n = 12 so we plug those in:
Sn = a1(1 - rn)/(1 - r)
S12 = (1)[1 - (1/2)12]/(1 - 1/2)
S12 = (1)(1 - 1/212)/(1/2)
S12 = (1 - 1/212)/(1/2)
S12 = (1 - 1/212)×(2/1)
S12 = 2(1 - 1/212)
S12 = 2 - 2/212
S12 = 2 - 1/211
S12 = 2 - 1/2048 = 1.9995
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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?
Well, we found:
S10 = 2 - 1/512 = 1.9980
S12 = 2 - 1/2048 = 1.9995
It appears that each time we will be
subtracting a smaller fraction away
from 2, so the whole number that the
it appears the sum will always be a
little smaller than -- is 2.
Edwin