In a geometric sequence, each new term is the product of the previous term and a fixed multiplier #r#. (This is in contrast to an arithmetic sequence, where each new term is the sum of the previous term and a fixed additive value #a#.)
To find the multiplier #r# of a geometric sequence, simply take the ratio of any two consecutive terms already in the sequence.
#r=(a_n)/(a_(n-1))#
For example, if we take the ratio of the 2nd-to-1st terms, we get:
#r=(a_2)/(a_1)="-14"/2="-7"#
Then, since we know the 4th term is -686, all we need to do is multiply this by our ratio #r# to get the 5th term:
#a_5=ra_4=("-7")("-686")=4802#.
Bonus:
In general, a geometric sequence is written as
#a,ar,ar^2,ar^3,...#
where #a# is the starting value of the sequence and #r# is the common ratio between successive terms. If you know both the first term and the common ratio, you can find the #n^"th"# term (#a_n#) by solving #a_n=ar^(n-1).# (In this case, we would get
#a_5=ar^(5-1)=2*("-7")^4=2*2401=4802,#
same as before.)
