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 the amount he paid the month before. Explain to Larry how to represent his first 15 payments in sequence notation. Then explain how to find the sum of his first 15 payments, using complete sentences.

f(n) = 150*1.3^(n-1), where n is the number of months
The sequence notation can be written as:
Pn = 1.3*P(n-1) for n > 1,  and P(1) = 150,
where Pn is the nth payment amount

To find the sum of the entire sequence, let's find a convenient summation notation:
Instead of n=1 to n=15, we're going to do n=0 to n=14, so that we can use some nifty properties of summation notation:

Sum of first 15 payments:
14
∑  150 * 1.3ⁿ  
0
Feel free to plug in a few values for n to confirm that the formula works. 

Pulling out the constant 150:
150 * 14                This is equal to 150 * (1 - 1.3^(n+1))÷ (1-1.3)) 
          ∑  1.3ⁿ  
          0


= 150 * (1-1.3¹⁵) ÷ (1.3) = 150 * 167.2863 = 25092.95