How do you use summation notation to expression the sum #0.1+0.4+1.6+...+102.4#?

1 Answer
Apr 15, 2018

Answer:

#sum_(n=0)^5 0.1(4)^n#

Explanation:

Judging by the first three terms, they don't share a common difference, so it will not be an arithmetic series; however, we should test the ratio of the terms to determine whether we have a geometric series:

#0.4/0.1=4#

#1.6/0.4=4#

So, we have a geometric series with the first term #a=0.1,# and common ratio #r=4.# This means that the #n#th term in the series is the first term #0.1# multiplied by #4^n, n>=0#.

Since the series ends at #102.4,# and we want to determine how many #n# we have, we'll solve this:

#0.1(4^n)=102.4#

#4^n=1024#

#n=5#

So, the series starts at #n=0,# ends at #n=5,# and the terms are given by #0.1(4)^n#:

#sum_(n=0)^5 0.1(4)^n#