How do you find the number of terms in the following geometric series: 100 + 99 + 98.01 + ... + 36.97?

1 Answer
Feb 27, 2016

Answer:

Find the general form of the #n^"th"# term of a geometric series and solve for #n# when that term of #36.97# to find that there are #100# terms in the given series.

Explanation:

A geometric series is a series of the form
#a_0+a_0r+a_0r^2+a_0r^3+...#
where #a_0# is the initial term and #r# is the common ratio between terms. We can easily find #r# by dividing any term after the first by the prior term. So in this case we have

#a_0=100# and #r = 99/100=0.99#

Now, looking at the general form of the series, we can see that the #n^"th"# term has the form #a_0r^(n-1)#. Thus, as we have the last term in the series, we simply need to solve for #n# for that term to find the total number of terms.

#36.97=a_0r^(n-1)=100(0.99)^(n-1)#

#=>0.3697=0.99^(n-1)#

#=>ln(0.3697)=ln(0.99^(n-1)) = (n-1)ln(0.99)#

#=>n=ln(0.3697)/(ln(0.99))+1 ~~ 100#

Thus there are #100# terms in the series.