Hi Jessica.

In a geometric series, consecutive terms are derived by multiplying the previous term by some constant factor. So that means the ratio between adjacent terms is a constant. In your case, -18/54 = -1/3, so
the common factor is -1/3.

Each geometric series starts with an initial value which we can call A, and has a common factor we can call F. That means the second value is AF, the third value is AF², the fourth value is AF³ and so on. In general, the "n"th value is AF^n-1.

Your sequence has F = -1/3, but what is A? You know that the 15th term is 54, so 54 = A(-1/3)¹⁴.

Thus A = 54(-3)¹⁴.

Now if you add up the second through fifth terms, you have AF + AF² + AF³ + AF⁴

which is just A (F + F² + F³ + F⁴).

Now you can substitute in your values of A and F to solve your problem.

Hope this helps,
Stephen La Rocque.