Question

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

Answer 1

`-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`