How do you find the fifth term in a geometric sequence in which the fourth term is 5, the sixth term is 7, and the common ratio is negative?

2 Answers
Nov 29, 2015

Answer:

#r= -sqrt("7th term"/"5th term")#

Explanation:

#r=-sqrt(7/5)#

5th term #=5xx-sqrt(7/5)~~-5.916#

hope that helped

Nov 29, 2015

Answer:

The fifth term will be a geometric mean of the fourth and sixth term.

Since the common ratio is negative it will be #-sqrt(35)#

Explanation:

The general form of a term of a geometric sequence is:

#a^n = a r^(n-1)#

Where #a# is the initial term and #r# the common ratio.

We are given #a_4 = a r^3 = 5# and #a_6 = a r^5 = 7#

So #r^2 = (a r^5) / (a r^3) = a_6 / a_4 = 7 / 5#

So #r = -sqrt(7/5)#

Then #a_5 = r a_4 = 5 (-sqrt(7/5)) = -sqrt(7)sqrt(5) = -sqrt(35)#

Or just taking the geometric mean of #a_4# and #a_6#...

#a_5 = +-sqrt(a_4 * a_6) = +-sqrt(35)#

and we need to use the negative square root to get a negative common ratio.