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.)
