On this problem, I mainly focussed on the second question, so that is what my solution is for.
Firstly, I noticed that the pattern was squares made up of more squares, but with the top right corner missing of the big square. This meant that each shape in the sequence was a square number minus 1.
The next thing that I worked out was the total number of smaller squares needed to make the shape. I realised that the side lengths of the shapes were only odd numbers:
3, 5, 7, 9, 11...
And the square numbers of them are:
9, 25, 49, 81, 121...
Then minus 1 for the missing square in the corner:
8, 24, 48, 80, 120...
I had now figured out the sequence, but in the form of numbers. I used this to find an equation to find out the number of small squares each part of the sequence contained. I firstly figured out that it was a quadratic equation, so it would be xn squared. I found out that the x would stand for 4 in this case, so it would be 4n squared.
I then set out to find what the last part of the equation was. As I knew that the 1st part of it was 4n squared, I found out what that was for each term in the sequence:
4, 16, 36, 64, 100...
I then took this away from the number of squares in each shape:
4, 8, 12, 16, 20...
I used these numbers to figure out that the last part of the equation was +4n. I worked this out because there was a difference of 4 between each number, so that would make it +4n.
The equation therefore is:
4n squared +4n