What is the 8th term of the geometric sequence 4, -20, 100?

1 Answer
Aug 13, 2017

#-312500#

Explanation:

Whenever looking at sequences, you have to attempt to find a common factor between the members of that sequence, be it the difference between them, a common divisor, a common multiplier, etc. In this case, we can see that if we divide each member by the previous one, we get a the same number:

#100/-20=-5#

#-20/4=-5#

So our sequence, starting at 4 is represented by the following recursive equation:

#a(1)=4#
#a(n)=a(n-1)xx-5 " for " n>1#

This is fine if we like calculating each previous member in order to get to whatever number in the sequence we need, but there is a better way (specially if the question asks for the 15th member or something higher).

Let's look at each calculation as a full equation:

#a(2)=a(1)xx-5=4xx-5=-20#

#a(3)=a(2)xx-5=a(1)xx-5xx-5=4xx-5xx-5=100#

#a(4)=a(3)xx-5=a(2)xx-5xx-5=a(1)xx-5xx-5xx-5=4xx-5xx-5xx-5=-500#

Notice that at each step all we're doing is adding a multiplication by #-5#, so at step 4, if we put all the multiplications by #-5# together we have #-5^3#. Notice that the power is one number less than the step. This allows us to write the following linear equation in reference to the step:

#a(n)=4xx-5^(n-1)#

Also note that this would even allow you to calculate the first step as any number raised to #0# is equal to 1. Now we just plug 8 into that equation and get our result.

#a(8)=4xx-5^7#

#a(8)=4*-78125#

#a(8)=-312500#