The terms form an arithmetic sequence whose sum is . The terms in that order, form a geometric sequence. Find the sum of all possible values for .

Source: NCTM Mathematics Teacher, February 2006

**SOLUTION**

Let represent the common difference of the arithmetic sequence

In summary, the terms of the arithmetic sequence are

The terms of the geometric sequence are

By definition of geometric sequence

The factors of are , and from which we form the following sequences

Of these sequences only and are geometric sequences. Thus, all possible values for are and . Their sum equals .

**Answer**:

*Alternative solution
*

from Eq.

from Eq.

By definition of geometric sequence

Substitute the value of from Eq.

or

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