Question

Given two terms in a geometric sequence, find the 8th term and the recursive formula.

Answer 1

Answer:

Step-by-step explanation:

Finding the 8th term of this sequence will be much easier with an explicit formula, so we will find that, find the 8th term, then write the equation for the recursive. The recursive formula is only good if you know all the terms before the one you're looking for. That means if we want the 8th term, we first have to find the 7th term, but before that we have to find the 6th term, etc...

We can find the explicit formula for this using the info we have about the 5th term and the 3rd term. The explicit formula looks like this, in general terms:

$a_n=a_1*r^{n-1$ where a1 is the first term and r is the common ratio. We can solve for both, one at a time. First, using the fact that the 5th term is 3888:

$a_5=a_1*r^{5-1$ and subbing in and simplifying:

$3888=a_1*r^4$ and we will solve that for a1:

$a_1=\frac{3888}{r^4}$

Now the other equation we need for this system, using the fact that the third term is 108:

$108=a_1*r^2$ and we sub in our expression for a1:

$108=(\frac{3888}{r^4})*r^2$ which simplifies to

$108=\frac{3888}{r^2}$ and

$r^2=\frac{3888}{108}$ so

r = 6. Now we'll plug 6 in for r to find a1:

$a_1=\frac{3888}{6^4}$ so

a1 = 3. Now we have the a and r for the explicit formula:

$a_n=3*6^{n-1$ and we simply put in an  for n:

$a_8=3*6^{8-1$ and

$a_8=3*6^7$ which gives us the 8th term as

a8 = 839808

The recursive formula is

$a_n=a_{n-1}*r$ and it works, also, but again, you have to have all the terms that come before the one you want or this formula is useless. Explicit is always better, in my opinion.