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