16+16+12=44
132÷44=3
3 are equal to the width.
So far we have w (2n + 12).
To see if the formula w (2n+12) works, we will try it out using the rose coloured box.
42+43+44+52+53+54= 288.
Width= 3.
3(2(42) +12): 3(84+12): 3(96).
3x96=288.
Once again, 76+77+78+86+87+88=492.
Width=3.
3(2(76) +12): 3(152+12): 3(164)
3x164=492.
This formula so far works.
But what would happen if the width changed?
This time, the width of each shaded square is 4.
For the first shaded square, the values come up to 180.
4(2n+12): 4(44): 4x44= 176.
As you can see, we have changed the width, but have not changed the n+12 to n+13. If we tried this, we would get 180.
To do this so our formula is correct we would have to alter it.
W (2n+r). ‘R’ stands for the range. To make sure this formula is correct, I will try it on the remaining two squares and then on one with a completely different width.
The sum of the rose coloured box comes up to 380.
W ((h)n+r)
4(2n+13): 4(82+13); 4(95)
4x95=380.
This answer is correct.
The value of the blue rectangle comes up to 660.
W (2n+r)
4(2n+13): 4(152+13): 4(165)
4x165= 660.
This time, the values add up to 70.
w(2n+r)
4(2n+3): 4(32+3): 4(35)
4x35=140. As you can see, the formula we used for the rectangles where the 2, the formula w(2n+r) worked.
16+17+18+19=70.
w(h(n)+r)
4(1(16)+3): 4(19): 4x19=76.
This answer is not correct so there is a problem in the formula.
1+2+3+4+11+12+13+14=60.
r(lxW)+(lxW)
2
13(2x4):13(8) =104.
104÷2=52
52+( LxW)
52+8=60.
r(LxW)+(lxW)
2
41(5x2)+(10)
2
41(10)+(10)
2
410÷2=205+10=215.
1+2+11+12+21+22+31+32+41+42=215.
This answer is correct, but would it still be if I changed the length.
All these numbers add up to 575.
R(LxW)+(LxW)
2
44(5x5):44(25): 1100. 1100÷2=550.
550+(LxW): 550+25=575.
This answer is correct, but what if we change the height.
All these numbers add up to 80.
R(LxW)+(LxW)
2
14(2x5)+(2x5)
2
14(10): 140. 140÷2=70. 70+(10)=80. So far this formula has been correct, but would happen if initial term changes from 1 to a different number.
R(LxW)+(LxW)
2
12(2x3)+(2x3)
2
12(6)=72. 72÷2=36. 36+6=42.
This answer is incorrect.
I could now try to split the height from 2 to 1 and adding both rows to get the final product.
w(in+1). I can change in which is the initial term to n, which will stand for the lowest number.
Width(n+1). 3(4+1)=3(5)=15. 4+5+6=15.
For the second row we will have 3(14+1)=3(15)= 45. 14+15+16=45.
45+15+60.
4+5+6+14+15+16=60.
54+55+56+57+64+65+66+67=484.
4(54+1) 4(55)= 220.
4(64+1) 4(65) 195. 220+260=480. This answer is incorrect. So instead of it being w(n+1), I am going to change it to w(n+(r-1)) for each row.
4 (54+(2))= 4(56)= 224
4(64+2)=4(66)=264. 224+264= 488. this answer is also incorrect.
