sequences and series - Blocks of Pyramid Pattern Expression

Wednesday 25 October 2017

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

Each layer (from the top).

Diagram.

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

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

Any ideas??

Thank you

Answer

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 $n+1$.

Blog