Question 1096056
<br>You didn't say what you are supposed to do with this pattern....  I would guess you are looking for a formula for the number of dots in the n-th figure.  Let's call the formula for the number of dots in the n-th figure D(n).<br>
For n=1, the pattern is a single dot.  D(1) = 1.
For n=2, the pattern is a 2x2 square, with 1 dot on top.  D(2) = 2^2+1 = 4+1 = 5.
For n=3, the pattern is a 3x3 square, with (2+1) dots on top.  D(3) = 3^2+(1+2) = 9+3 = 12.
For n=4, the pattern is a 4x4 square, with (3+2+1) dots on top.  D(4) = 4^2+(1+2+3) = 16+6 = 22.<br>
So there are two parts to the pattern.  We always have an nxn square of dots; and above that we have a number of dots which is the sum of the whole numbers from 1 to (n-1).<br>
You might be familiar with the formula for the sum of the integers from 1 to n; it is
{{{n(n+1)/2)}}}<br>
So the sum fo the integers from 1 to (n-1) is
{{{(n-1)(n)/2}}}<br>
And so the number of dots in the n-th figure is
{{{n^2 + (n-1)(n)/2 = n^2 + n^2/2 - n/2 = (3n^2-n)/2 = n(3n-1)/2}}}