Math Question - See "Add Details" below - no guesses show working to prove solutions?

Question

A perpetuity costs 77.1 and makes annual payments at the end of the year. The perpetuity pays 1 at the end of year 2, 2 at the end of year 3, …., n at the end of year (n+1). After year (n+1), the payments remain constant at n. The annual effective interest rate is 10.5%. Calculate n.

Answer 1

n = 19 (The value at that point (after payment) is 180.9766 and it rises .0026-ish each year after but raising n to equal 20 makes it fall off rapidly so this is the sweet spot described in the problem.)

But... I solved it with a spreadsheet and brute force so...

So far anyhow. We'll see if I have time to do better.


Added:

disposable_hero_too has a nice approach except for one thing: the payouts. No further amount is earned on the payouts and they delay reaching the stable point desired. To n = 19 in fact (20th year).

Answer 2

The growth in any year n is defined by the formula n(n-1)/2.

This is derived from the formula for the sum of consecutive integers which is: n(n+1)/2. However, since n represents the number of years, but interest doesn't accrue until year 2, we will substitute n-1 for n: (n-1)(n-1 + 1)/2 = (n-1)(n)/2 or (n^2 - n)/2.

So our total value will be 77.1 + (n^2 - n)/2 in any given year (initial plus growth).

We want to set 77.1 + (n^2 - n)/2 equal to a an equivalent growth rate of 10.5% at year n.

So...

(n^2 -n)/2 / 77.1 / n = 0.105
n^2 - n = 0.105 * 77.1 * 2 *n
n^2 - n = 16.191n
n^2 - 17.191n = 0

we know n cannot be zero, so n = 17.191, or 17 years.

Hope this helps! :)