Now that I have got my shapes I am going to make a table to record there amount off boxes in the pyramid and also to find out there first and second differences. The reason I’m going to do differences is so I can find out how many boxes are between each pyramids, also I’m doing this because I feel it might be useful later on when I try and find out the formula.
2)
Now that I have the first and second differences I am going to find the nth term. I am going to do this by using quadratics because I find it is the easiest way to do so. And by using quadratics I can also use the first and second differences to my advantage because now I can add the differences in to the quadratic function, also I can easily find out the nth terms of the differences using the function.
3)
C=
1st B=
2nd A=
2a=1 a+b=1 c=0
a=0.5 b=0.5
The line is here to separate the differences from the nth term
As we can see here in the quadratic equation the second differences are constant and we have got all the nth terms.
Next I am going to give a certain part of the pyramid a letter so that I can use that letter in the formula.
4)
X= The number of blocks on the bottom of the pyramid
Y= The number of boxes in the pyramid
Now that I have done that I will now move on to making the formula. I will use the nth terms and there letters and I will also use the letters that I have given to the pyramid.
Y=ax2+bx+c
Now that I have what I think is the formula I am going to do a few examples first to make sure that the formula that I have is the right one.
X=1 Y=1
Y=a * 12+b * 1+c = 1
X=2 Y=3
Y=a * 22+b * 2+c = 3
X=3 Y=6
Y=a * 32+b * 3+c = 6
X=4 Y=10
Y=a * 42+b * 4+c = 10
What I have done here is added X and Y too the formula so that the formula is complete. When the numbers from X and Y where added I worked the formula so that in the end I would get the number of boxes in the pyramid. When I thought I had the answer I checked the pictures on pages 3,4,5 to make sure that the number was correct. As you can see the formula has worked out and is working.
Y=1/2X2+1/2x+c
This is the formula used to find the number of blocks in a pyramid when given one number.
Introduction.
In this section I am going to do the same as I done in the first section but this time I’m going to use my own shape. I will have to work out a formula for a new shape and I’m going to do this in the same way as I did in part 1 but this time I’m going to use squares instead of pyramids.
- Here I’m going to draw a number of shapes so I can get a pattern or a list of numbers, which go with the shape.
This square is on its own and doesn’t need to support anything else.
This square has four blocks two on the bottom and two on top of them.
This square has a top, middle and bottom row each of them has three blocks which all adds up to nine blocks.
This square is almost the same as the square before except it an extra row and column of four.
This is the last and the biggest square it has twenty-five blocks, which are in rows of five.
Now that I have got all my shapes and the number of blocks in them shapes I will put them into a table. I will use the table to find the first and second differences. The reason for this so I am able to see how many blocks there is between the squares also I am able to use the differences later on when it comes down to finding out the formula.
2)
Now that I have found the first and second differences I will use the numbers that I have to find the nth term by using quadratics. By using quadratics I am able to use the differences. The object of me using the quadratics is to find the nth terms of the differences and then use the nth terms to get a,b,c, values that will be used in the formula.
3)
2a=2 a+b=1 c=0
a=1 b=0
This line is here to separate the differences from the nth terms.
Now that I have the nth terms I am able to make a formula. Here I am going to give different parts of the square a letter so that when it comes to making the formula I don’t have to write down which part it is.
R= The number of blocks on the bottom row.
N= The number of boxes in the cube.
Now that I have this done I am able to make a formula. Here I am going to make the formula, which will tell me how many blocks are in a square when I put in a number.
N2=ar+br+c
Now that I have a formula for my problem all I have to do now is to try it out to make sure it works. So I am going to give my formula a set of numbers and see if it gives me the right answer.
R=1 N=1
N2=a * 1+b * 1+c=1
R=2 N=4
N2=a * 2+b * 2+c=4
R=3 N=9
N2=a * 3+b * 3+c=9
R=4 N=16
N2=a * 4+b * 4+c=16
From the above I can see that my formula works and that it gives me the right number of blocks out when I put a certain number in. When the numbers from X and Y where added I worked the formula so that in the end I would get the number of boxes in the square. When I thought I had the answer I checked the pictures on pages 11,12 to make sure that they where correct.
This is the formula that when you put in a number it will give you the number of blocks in the square.
N2=ar+br+c
