Trying to put this idea into a formula

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mdw

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I am trying to figure out a formula to describe the following scenario:


If I do a nice favor for 1 person a week, and ask each of them to do the same, how many people would be on any given level on any given week, assuming that they all do a kind favor for 1 person a week?


For example, the first week, there is one person on "Level" one. (This is one person that I personally did a favor for; Let's call him Tom.) The next week, there are 2 people on "Level One" and one person on "level Two", because I did another favor for a different person (Let's call her Sally.) and Tom also did a kind favor.

Please help!
 
\(\displaystyle 1 + 2 + 3 + ... + n = \frac{n\cdot(n+1)}{2}\)
 
Is "n" the week or the level?

Let's say that n=5, 1+2+3+4+5 = (5(5+1))/2
so, 15 = 15 (Huh?! Now I'm even more confused!)

Hold on. 15 = 3n/2
10 = n

I still don't know what this really means in the scenario described. :/ Please help.
 
Make charts and look for patterns,
Level 1's:
week 1: 1
week 2: 2
week 3: ?
week 4: ?

then do the same for level 2's

Level 2's
week 1: ?
week 2: ?
week 3: ?
week 4: ?

I don't think n(n+1)/2 really helps here.
 
Yes, I've done what you said, and I thought that I'd found a formula that worked:

Level(N) = Week # -(N-1)

But that only works up until Week 3, so then I tried:

Level (N) = Week # - (N-1) + (total from previous week on same level)

But that only works until level 4.


Week 1
Level 1: 1

Week 2
L(1) = 2
L(2) = 1

Week 3
L(1) = 3
L(2) = 3
L(3) = 1

Week 4
L(1) = 4
L(2) = 6
L(3) = 3
L(4) = 1
 
Just looking at the level ones, I interpret the question like so:

Week 1:
Level 1: 1

Week 2:
Level 1: 2
Level 2: 1

At this point doesn't everyone involved create a level 1? That's 4 people including "you". The 2 level 1's become level 2's.
Based on how the question is worded I am not counting "you" (the original favor giver) as having a level.

Week 3:
Level 1: 4
Level 2: 2
Level 3: 1

If it turns out like this the formula turns out to be pretty simple.
 
Okay, to answer your question first, yes, everyone creates a level 1 (even people on my level 2,3,4, and 5) on their second week of involvement. And, yes, you are correct, I'm not including "you" on any level.

You stated "The 2 level 1's become level 2's." That is incorrect. They create level 1's on their second week of involvement.

I have sat down and done the first few levels by hand and the levels that I input before are the correct numbers. I just can't draw out each level; it gets to be messy and time-consuming. That's why I'm looking for a formula.

Maybe I can re-word it to more accurately convey my ideas.

On week 1, "YOU" do a favor for 1 other person. Every week, every one involved (including yourself) does a favor for one other, new person. For any given week (Wk #), how many total people are on any given level (L(n))?
 
Week 1
Level 1: A

Week 2
L(1) = A,B = 2
L(2) = A1 = 1 On wk2, the 1st person (A) does their first favor for person (A1).

Week 3
L(1) = A,B,C = 3
L(2) = A1, A2, B1= 3 On wk 3, (A) & (B) do a favor for a new person, (A2) & (B1) respectively.
L(3) = A1.1=1 On wk 3, (A1) does a favor for the first time for (A1.1), creating L(3).

Week 4
L(1) = A,B,C,D = 4
L(2) = A1, A2, A3, B1, B2, C1 = 6
L(3) = A1.1, A1.2, A2.1, B1.1 = 4
L(4) = A1.1-1 = 1
 
Last edited:
Is excel an option? It seems pretty recursive and is something it could handle extremely easily. Also I think week 4, L(3)=4.
 
Yes, you are correct. Wk4, L(3) = 4

And, yes! I would love to put it in excel. In fact, that's what I intend to do, but I have to give excel a formula, right?

Thx for all your help thus far!
 
Level 1 : 1 2 3 4 5
Level 2 : 0 1 3 6 10
Level 3 : 0 0 1 4 10
Level 4 : 0 0 0 1 5
Level 5 : 0 0 0 0 1

The columns are the weeks 1-5

Just type in 1 2 in the first row, select both of those cells and drag it to the right as far as you want.
Type 0 in 2nd row. Then formula is just cell B2=A1+B1. Then drag this cell all the way right. then select that whole row and drag it down as far as you want.
 
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