What will be the base case here in this recurrence relation?

Question

Problem:

Lorenzo takes up a loan of 40,000. It is to be paid by annual installments of 2000 with first payment made at the end of the first year the loan was taken out. 3% interest is charged at the end of each year on the remaining debt. Model an appropriate recurrence relation for the remaining debt. Determine a solution to the recurrence relation.

My question:

What is the value of $a_1$ here? That is, what will be the debt at the end of the first year? Will it be $40000-2000=38000$ or will it be $40000 - 2000 + (40000-2000)\cdot\frac{3}{100}=39140$?

Your second option should be modified to read $(40000-2000)*(1+\frac 3{100})$. — abiessu, Mar 04 '23 at 15:02
In other words, "the interest is charged on the remaining debt" means that $38000$ gets $3$% added to its value. — abiessu, Mar 04 '23 at 15:04
$a_1=38000$ according to the words in the problem, but is probably expected to be $39140.$ — abiessu, Mar 04 '23 at 15:07
Either way, you can solve the problem correctly, you just have to be precise on what you mean by each term you use. — abiessu, Mar 04 '23 at 15:08
The problem seems to be worded in a way to increase confusion rather than make it easier... — abiessu, Mar 04 '23 at 15:11

score 1 · Accepted Answer · answered Mar 05 '23 at 05:37

The wording in this problem allows for two scenarios: either the payment is applied to the debt first at the end of the year, or the interest is applied first at the end of the year. The first case matches the order in which the events are described, and is the likely scenario, giving $a_1=38000$ as the "remaining debt" to which the interest is applied. The alternate scenario where the interest happens first gives $a_1=40000$ as the "remaining debt."

This alternate scenario is only worth discussing in the pessimistic mindset that the entity loaning the money gets more back this way.

Comparing the results after the first year and all events have occurred, we get $39140$ in debt for the first case vs. $39200$ for the second. This disparity grows as the payments continue, reaching $1200$ just from making $20$ payments (not accounting for compounding).

What will be the base case here in this recurrence relation?

1 Answers1