**TI Nspire CX II: Distributing Money Among Friends**

**The Problem**

This post is inspired from a Instagram post by @gercekboss at www.eylemmath.weebly.com. The problem states that:

*$100 is to be distributed among 100 friends:*

*1% is given to the 1st friend*

*2% is given to the 2nd friend*

*3% is given to the 3rd friend*

*and so on.*

I have a TI Nspire CX II Document that covers this problem, which can be downloaded here: https://drive.google.com/file/d/1aewZ-8wU2JjZxgGyBpNHivEYLU5QAt31/view?usp=sharing

Name of the document: distribute percent.tns

Using the TI Nspire CX II to set up two sequences to model this problem as two sequences:

**Balance Sequence (Problem 1.3, plot is in green)**

u1(n) = u1(n-1) - u1(n-1) * n%

Intial Terms: 100

0 ≤ n ≤ 99 step 1

**Amount Sequence (Problem Page 1.7, plot is in red)**

u2(n) = u1(n-1) *n%

Initial Terms: 0

0 ≤ n ≤ 99 step 1

By analysis, the 10th friend will get the biggest distribution under this plan, $6.28. By the time we get to the 38th friends, the distribution becomes meaningless because the calculated amount will be less than one penny.

**Regression Analysis**

I performed curve fitting analysis for the balance and amount distributed on problems page 1.5 and 1.9, respectively.

Balance can be estimated by the logistic equation:

balance(n) ≈ 112.84/(1 + 0.1*e^(0.23n)) - 0.54 for 1 ≤ n ≤ 40

The maximum absolute error is 0.67.

Amount can be estimated by the quartic polynomial:

amount(n) ≈ -4.01E-5*n^4 + 4.31E-3 * n^3 - 0.15 * n^2 + 1.89 * n - 1.26

for 1 ≤ n ≤ 40

I did slightly better, the maximum absolute error is 0.52.

**Generalizing the Problem**

Now, let's take any money of money and any amount of friends for whom to distribute that money. The rate can increase per friend by at different rates:

a*n% + b%

If you want 2%, 4%, 6%, etc, let a = 2, b = 0

If you want each friend to get 1/2% more than the last, let a = 1/2, b = 0.

If you want the first friend to get 2%, second to get 3%, etc, let a = 1, b = 1.

The sequences are set up as:

**Balance Sequence**

u1(n) = u1(n-1) - u1(n-1) * (a*n% + b%)

Intial Terms: c

0 ≤ n ≤ l step 1

**Amount Sequence**

u2(n) = u1(n-1) *(a*n% + b%)

Initial Terms: 0

0 ≤ n ≤ l step 1

Variables:

a = rate parameter

b = rate parameter

c = initial amount of money

l = number of friends

You can customize the problem in Problem 2 of the document. Change the variables on Problem Page 2.2 and see the results on Problem Pages 2.3 and 2.4.

Source:

Question 251. eleymmath. https://eylemmath.weebly.com/algebra/100-question Retrieved July 21, 2019.

Eddie

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