106k views
4 votes
If the second term of a geometric sequence of real numbers is -2 and the fifth term is 16 then what is the fourteenth term?

1 Answer

1 vote

Given

A geometric sequence such that ...


a_2=-2\quad\text{and}\quad a_5=16

Find


a_(14)

Solution

We can use the ratio of the given terms to find the common ratio of the sequence, then use that to find the desired term from one of the given terms. We don't actually need the common ratio (-2). All we need is its cube (-8).


a_2=a_1r^((2-1))=a_1r^1\\a_5=a_1r^((5-1))=a_1r^4\\a_(14)=a_1r^((14-1))=a_1r^13=a_5r^9\\\\(a_5)/(a_2)=(a_1r^4)/(a_1r^1)=r^3=(16)/(-2)=-8\\\\r^9=\left(r^3\right)^3=(-8)^3=-512\\a_(14)=a_5(-512)\\\\a_(14)=-8192

User Atoms
by
6.3k points