Jeff's college loan repayment strategy follows a consistent pattern. He starts with an initial payment of $175 and increases his subsequent payments by a factor of 1.4 each month. This creates a geometric sequence, where each term is found by multiplying the previous term by a constant value (in this case, 1.4). This p...
Representing Jeff's Payments in Sigma Notation
Sigma notation provides a concise way to represent the sum of a series. For Jeff's payments, we can express the sum of his first 36 payments as:
∑_(i=1)^36 175 * (1.4)^(i-1)
Let's break down this notation:
- ∑: This symbol represents the sum of a series.
- (i=1): This indicates the starting value of the index 'i', which represents the payment number. The first payment is numbered '1'.
- 36: This is the upper limit of the index, meaning we sum up to the 36th payment.
- 175 * (1.4)^(i-1): This is the formula for each individual payment. The base payment of $175 is multiplied by 1.4 raised to the power of (i-1). This ensures each payment is 1.4 times greater than the previous one.
Calculating the Total Amount Paid
To find the total amount Jeff pays over 36 months, we need to calculate the sum of the geometric series represented by the sigma notation. The formula for the sum of a finite geometric series is:
S_n = a(1 - r^n) / (1 - r)
Where:
- S_n: The sum of the first 'n' terms of the series
- a: The first term of the series (Jeff's initial payment of $175)
- r: The common ratio (1.4 in this case)
- n: The number of terms (36 payments)
Plugging in the values:
S_36 = 175(1 - 1.4^36) / (1 - 1.4)
Calculating this expression, we find:
S_36 ≈ $310,793.25
Therefore, Jeff would pay approximately $310,793.25 in total over 36 months.
Determining Convergence or Divergence
A geometric series converges if the absolute value of the common ratio (|r|) is less than 1. If |r| is greater than or equal to 1, the series diverges. In Jeff's case, r = 1.4, and |1.4| > 1. This means the series is **divergent**. A divergent series does not have a finite sum. This implies that Jeff's loan payments will continue to grow exponentially, making it increasingly difficult for him to manage his debt.
While this analysis provides a clear picture of Jeff's payment trajectory, it's important to note that real-world loan scenarios are more complex. Interest rates, loan terms, and other factors influence the actual repayment process. This analysis serves as a simplified model to illustrate the concept of geometric series and its implications for understanding loan payments.