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There is a pattern following, and trying to find the algebraic expression

Each layer (from the top).


enter image description here

So the first layer has 1, second has 4, third has 9, and the fourth has 16.

That's how the sequence is increasing.

What I'm looking for is,

When the second layer is added with the first layer,

Third layer is added with the second and first,

Fourth is added with third,second and first.

So something like this.

enter image description here

enter image description here

I am trying to find the algebraic expression for this pattern.

Any ideas??

Thank you


marked as duplicate by Arthur, user147263, Ivo Terek, Joel Reyes Noche, Rick Decker Dec 18 '14 at 1:02

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


There is a well-known formula for the sum of the first $n$ squares, but I don't want spoil your investigation, so I will give you some hints.

First, compute some more terms of the sequence. Three or four more should do.

Multiply all the terms by six, and factor the results. Notice that all of them are multiple of $n$ and $n+1$.

  • $\begingroup$ Why do I multiply by 6? $\endgroup$ – didgocks Dec 17 '14 at 23:34
  • $\begingroup$ Beacause, erm, it is easier that way :) $\endgroup$ – ajotatxe Dec 17 '14 at 23:35
  • $\begingroup$ Were do I get (2n+1) from? $\endgroup$ – didgocks Dec 17 '14 at 23:40

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