# Growing Squares

John McLaughlin –

Feel free to e-mail me on ways to improve this essay it would be a great help.

## Growing Squares

I have decided to find a formula to find the nth term.  To help me find the nth term I shall compose a table including all the results I know.

The 2nd difference is constant; therefore the equations will be quadratic.  The general formula for a quadratic equation is an2 + bn +c.  The coefficient of n2 is half that of the second difference

Therefore so far my formula is: 2n2 + [extra bit]

I will now attempt to find the extra bit for this formula.

From my table of results I have found the formula to be 2n2 + 2n + 1

I will now check my formula by substituting a value from the table in to my formula:

E.g. n = 2

Un = 2 (2) 2 – 2 (2) + 1 = 8.

For Diagrams 1 – 4 I can see a pattern with square numbers.  The diagrams numbers squared added to one less than the diagram number squared gives the correct number of squares.

For diagram n it should be:

Un = (n – 1) 2 + n2

(n – 1)(n – 1) + n2

n2 – n – n – n + 1 + n

Un = 2n2 – 2n + 1

This is correct.

Growing Hexagons

I will now repeat my investigation, and change the original shape of the square to hexagons, and try to find the formula as before.

I shall start by finding the width of each hexagon.

I have found the 1st difference to be constant; therefore the formula will be linear.

I have found the formula to be W=2n-1

I will now try to find the number of hexagons with in a pattern

Table of results:

The 2nd difference is constant; therefore the ...