Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -36 and 2304, respectively.

Answer 1

When we find a geometric sequence using the first and second terms, what we do is divide the second by the first to find a "common ratio," the number you multiply by to get to the next term.

In this case, we only have the second and fifth terms. From the second to the fifth term, we've multiplied by that common ratio three times.
What we can do is divide the fifth by the second and then take the cube root to find our common ratio.

$2304\div-36=-64\\\sqrt[3]-64=\boxed{-4}$

Now we know we're multiplying by -4 each time, let's find the first term.

The opposite of multiplying by -4 would be dividing by -4. We work back from our second term -36 to find that the first term would be 9.

For each sucessive term, we then multiply by -4.

Our equation is $a_n=9\times-4^{(n-1)}$