Larry has taken out a loan for college. he started paying off the loan with a first payment of $150. each month he pays, he wants to pay back 1.3 times as the amount he paid the month before. explain to larry how to represent his first 15 payments in sigma notation. then explain how to find the sum of his first 15 payments, using complete sentences. explain why this series is convergent or divergent.

This is definitely a case of geometric progressions.
The first value (first payment) is $150.  The next is $150*1.3; the next is $150*1.3*1.3.  And so on.  The formula used to determine the nth payment is 

Pn = P*r^(n-1), where P is the first payment and n is the index.  For the first payment, n=1 and the predicted payment is $150*1.3^(1-1) = $150, as expected.
 

We want to find the sum of this person's first 15 payments.  The formula for the sum of the first n payments is
            
           1-r^n
Sn = , and r = 1.3 in this case and n=15.
             1-r
                          1 - 1.3^15
Thus, S_15 =
                             1-1.3