What is the 5th term in the following geometric sequence 2, -14, 98, -686?

1 Answer
Dec 11, 2016

Answer:

#a_5=4802#.

Explanation:

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.)