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 Question from jessica, a student: If a geometric series includes 54-18+6-2 as its fifteenth through eighteenth terms, find the sum of the second through the fifth term, inclusive.

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 AFn-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.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.