Number Stairs

Which means that (6 x STAIR NUMBER) + 40 = STAIR TOTAL.

Now that I have worked out the formula for the 9x9 grid I am going to use the formula in random staircases with random stair numbers. And those are:

    Stair number = 61

    Stair total = (6x61) + 40 = 406, or alternatively,                 61+62+63+70+71+79= 406

Stair number=7

Stair total= (6x7) + 40= 82

With out the nth term the stair total= 7+8+9+16+17+25=82

Stair number=55

With out the nth term stair total= 55+56+57+64+65+73= 370

With nth term, stair total= (6x55) +40=370

Here is an alternative way to find the stair total of the 9x9 grid by using further algebraic method.

As we can see here n=stair number, and the 3x3 stair case from the 9x9 grid can be substituted in to the formula staircase for the 9x9 grid.

Total for algebraic staircase= n+n+1+n+2+n+9+n+10+n+18=  6n+40

We can also evaluate that Stair number (n) = 55

By substitution stair total= 55+55+1+55+2+55+9+55+10+55+18=370

Here is another example:

                                                                                                                                                                

By substitution, stair total= 7+7+1+7+2+7+9+7+10+7+18=82

Here is another example:        

It is apparent that the stair number = 39

By substitution stair total= 39+39+1+39+2+39+9+39+10+39+18=274

Now that I have done my further investigation for the 3 step stair for the 9x9 grid, now I am going to undertake further investigation for 3 step stair for the 8x8 grid.

Here is an 8x8 grid showing the stair total and the stair number:

                                                                                   Stair number (n)=1

By calculating the sum of all figures inside the stairs gives you the stair total. With the stair number 1 we get a stair total of 1+2+3+9+10+17= 42 =T. The stair total is calculated accordingly to the stair number for any grid size. This is for the 8x8 grid with the stair number 1.

                                                                                          Stair number= 2

Whereas the stair total= 2+3+4+10+11+18 = 48 = T

                                                                                        Stair number=3

Whereas the stair total= 3+4+5+11+12+19= 54 = T

 The following table shows the stair total (T) depending on the relevant stair number for the 8x8 grid. Which are from 1 to 5.

                         

42    48    54   60    66

                 +6                                  +6                                  +6                              +6

It is clear that the difference between the numbers is 6. This means that the formula, to solve the total, must include a multiple of 6. This will lead to the fact that the nth term has to have 6n in the formula, the extra is calculated by working out what is left over this will be +36.Therefore, the formula has to be T =6n + 36. By substituting the stair number to the nth term we get the stair total. Here we can see that is clearly evident that the remaining part in the nth term for the 8x8 grid has decreased by 4 as compared to the 9x9 grid.  The way how this formula works is the following:

Which means that (6 x STAIR NUMBER) + 40 = STAIR TOTAL

Now that I have worked out the formula for the 8x8 grid I am going to use the formula in random staircases with random stair numbers. And those are:

Here we can see that stair number= 46

Whereas stair total= 46+47+48+54+55+62=312

But by using the nth term, stair total= (6x46) + 36= 312

Stair number=41

Whereas stair total= 41+42+43+49+50+57= 282

But by using the nth term, stair total= (6x41) + 36= 282

Stair number=27

Whereas stair total= 27+28+29+35+36+43=198

But by using the nth term, stair total= (6x27) + 36= 198

Here is an alternative way to find the stair total of the 8x8 grid by using further algebraic method.

As we can see here n=stair number, and the 3x3 stair case from the 8x8 grid can be substituted in to the formula staircase for the 8x8 grid.

Total for algebraic staircase= n+n+1+n+2+n+8+n+9+n+16=  6n+36

We can also evaluate that Stair number (n) = 6

By substitution stair total= 6+6+1+6+2+6+8+6+9+6+16=72

Here is another example:

Here we can see that n= 46

By substitution, stair total= 46+46+1+46+2+46+8+46+9+46+16= 312

Now that I have done my further investigation for the 3 step stair for the 8x8 grid, now I am going to undertake further investigation for 3 step stair for the 7x7 grid.

Stair number=1

Whereas stair total= 1+2+3+8+9+15= 38

Here we can see that stair number=2

Therefore, stair total= 2+3+4+9+10+16=44

Here we can see that stair number=3

Therefore, stair total= 3+4+5+10+11+17= 50

Stair number (n) = 4

Therefore, stair total= 4+5+6+11+12+18 = 56

The following table shows the stair total (T) depending on the relevant stair number for the 7x7 grid. Which are from 1 to 5.

38    44    50   56    62

                    +6                               +6                             +6                                   +6

It is clear that the difference between the numbers is 6. This means that the formula, to solve the total, must include a multiple of 6. This will lead to the fact that the nth term has to have 6n in the formula, the extra is calculated by working out what is left over this will be +32.Therefore, the formula has to be T =6n + 32. By substituting the stair number to the nth term we get the stair total. Here we can see that is clearly evident that the remaining part in the nth term for the 7x7 grid has decreased by 4 as compared to the 8x8 grid.  The way how this formula works is the following:

Which means that (6 x STAIR NUMBER) + 32 = STAIR TOTAL

Now that I have worked out my nth term for the 3 step stair case in the 7x7 grid size, I am going to use this equation into random 3 step stair case in the 7x7 grid, in order to find out the stair total and to see if the formula works.

Stair number=33

Stair total= 33+34+35+40+41+47= 230

But by using the nth term our stair total= (6x33) + 32=230

Stair number = 29

Stair total= 29+30+31+36+37+43 = 206

By using nth term our stair total= (6x29) +32= 206

Stair number = 19

Stair total= 19+20+21+26+27+33= 146

By using our formula the stair total= (6x19) + 32= 146

Here is an alternative way to find the stair total of the 7x7 grid by using further algebraic method.

As we can see here n=stair number, and the 3x3 stair case from the 7x7 grid can be substituted in to the formula staircase for the 7x7 grid.

Total for algebraic staircase= n+n+1+n+2+n+7+n+8+n+14=  6n+32

We can also evaluate that Stair number (n) = 16

By substitution stair total= 16+16+1+16+2+16+7+16+8+16+14= 128

Here is another example:

Here we can see that n= 12

By substitution, stair total= 12+12+1+12+2+12+7+12+8+12+14= 104

Now I have done the investigation for the 3 step stair case for the 7x7, 8x8, 9x9, 10x10 grids. Now I am going to undertake my investigation for 4 step stair cases for the 77x, 8x8, 9x9, and 10x10 grids.

I will once again use (T) for my stair total, and (N) for my stair number.

I am going to start with the 10x10 grid.

                                                                                                    N= 1

The stair total is the value of the addition of all the figures inside any step stair cases. Hence, T = 1+2+3+4+11+12+13+21+22+31= 120 = Stair total.

Stair number=2

Stair total= 2+3+4+5+12+13+14+22+23+32= 130

Stair number=3

Stair total= 3+4+5+6+13+14+15+23+24+33= 140

Stair number= 4

Stair total = 4+5+6+7+14+15+16+24+25+34= 150

The following table shows the stair total (T) depending on the relevant stair number for the 10x10 grid, with stair case number4. Which are from 1 to 5.

120    130    140     150      160

                +10                                  +10                              +10                            +10

It is clear that the difference between the numbers is 10. This means that the formula, to solve the total, must include a multiple of 10. This will lead to the fact that the nth term has to have 10n in the formula, the extra is calculated by working out what is left over this will be +110.Therefore, the formula has to be T =10n + 110. By substituting the stair number to the nth term we get the stair total. The way how the formula works is the following:

Which means that (10 x STAIR NUMBER) + 110 = STAIR TOTAL

Now that I have worked out my nth term for the 4 step stair case in the 10x10 grid size, I am going to use this equation into random 4 step stair case in the 10x10 grid, in order to find out the stair total and to see if the formula works.

Stair number=57

Stair total = 57+58+59+60+67+68+69+77+78+87 = 680

But by using the nth term, stair total= (10x57) + 110 = 680

Stair number = 51

Stair total = 51+52+53+54+61+62+63+71+72+81 = 620

By using nth term stair total = (10x51) + 110 = 620

Stair number = 7

Stair total = 7+8+9+10+17+18+19+27+28+37 = 180

By using the nth term stair total = (10x7) + 110 = 180

Here is an alternative way to find the stair total of the 10x10 grid by using further algebraic method for the 4 step stair case.

As we can see here n=stair number, and the 4x4 stair case from the 10x10 grid can be substituted in to the formula staircase for the 10x10 grid.

Total for algebraic staircase= n+n+1+n+2+n+3+n+10+n+11+n+12+n+20+n+21+n+30 = 10n + 110

We can also evaluate that Stair number (n) = 36

By substitution stair total= 36+36+1+36+2+36+3+36+10+36+11+36+12+36+20+36+21+36+30= 470

Here is another example:

As we can see here n=stair number, and the 4x4 stair case from the 10x10 grid can be substituted in to the formula staircase for the 10x10 grid.

We can also evaluate that Stair number (n) = 17

By substitution stair total= 17+17+1+17+2+17+3+17+10+17+11+17+12+17+20+17+21+17+30= 280

Now that I have accomplished my investigation on the 10x10 grid with 4 step stair case, I am going to the 4 step stair investigation on the 9x9 grid.

                                       

                                                                      Stair number (N) =1, stair total=1+2+3+4+10+11+12+19+20+28= 110

Stair number (n) = 2

Stair Total = 2+3+4+5+11+12+13+20+21+29= 120.

Stair number = 3

Stair total = 3+4+5+6+12+13+14+21+22+30= 130

Stair number=4

Stair total= 4+5+6+7+13+14+15+22+23+31=140

The following table shows the stair total (T) depending on the relevant stair number for the 9x9 grid, with stair case number4. Which are from 1 to 5.

110    120    130     140      150

                       

                       +10                              +10                           +10                              +10

It is clear that the difference between the numbers is 10. This means that the formula, to solve the total, must include a multiple of 10. This will lead to the fact that the nth term has to have 10n in the formula, the extra is calculated by working out what is left over this will be +100.Therefore, the formula has to be T =10n + 100. By substituting the stair number to the nth term we get the stair total. The way how the formula was in algebraic terms is:

Formula to find out and nth term of any linear sequence = dn + (a-d) =T

a= First term in the linear sequence = 110

d= common difference = 10

n=stair number

T = stair total

By substitution, nth term was formed by: (10)(n) + (110-10) = 10n + 100

Now that I have figured out the formula for the 4 step stair for the 9x9 grid, I am going to test this formula in to random stair cases in the 9x9 grid.

Stair total = 51+52+53+54+60+61+62+69+70+18= 610

By formula, stair total = (10x51) + 100 = 610

Stair total = 46+47+48+49+55+56+57+64+65+73 = 560

By using nth term, stair total = (10x46) + 100= 560

Here is an alternative way to find the stair total of the 9x9 grid by using further algebraic method for the 4 step stair case.

As we can see here n=stair number, and the 4x4 stair case from the 9x9 grid can be substituted in to the formula staircase for the 9x9 grid.

Total for algebraic staircase= n+n+1+n+2+n+3+n+9+n+10+n+11+n+18+n+19+n+27 = 10n + 100

We can also evaluate that Stair number (n) = 24

By substitution stair total= 24+24+1+24+2+24+3+24+9+24+10+24+11+24+18+24+19+24+27= 340 = T

Now that I have accomplished my investigation on the 9x9 grid with 4 step stair case, I am going to the 4 step stair investigation on the 8x8 grid.

Here we can see that, stair number=1

Stair total= 1+2+3+4+9+10+11+17+18+25= 100

Here we can see that stair number = 2      Stair number = 3

Whereas, stair total = 2+3+4+5+10+11      Stair total = 3+4+5+6+11+12+13+19+20+27=120

+12+18+19+26= 110

The following table shows the stair total (T) depending on the relevant stair number for the 8x8 grid, with stair case number4. Which are from 1 to 5.

100    110    120     130      140

                       

                 +10                                  +10                               +10                           +10

Now that I solve the total, must include a multiple of 10. This will lead to the fact that the nth term has to have 10n in the formula, the extra is calculated by working out what is left over this will be +90.Therefore, the formula has to be T =10n + 90. By substituting the stair number to the nth term we get the stair total. The way how the formula was in algebraic terms is:

Formula to find out and nth term of any linear sequence = dn + (a-d) =T

a= First term in the linear sequence = 100

d= common difference = 10

n=stair number

T = stair total

By substitution, nth term was formed by: (10)(n) + (100-10) = 10n + 90

The way how the formula works is the following:

Which means that (10 x STAIR NUMBER) + 90 = STAIR TOTAL

Here is an alternative way to find the stair total of the 8x8 grid by using further algebraic method for the 4 step stair case.

As we can see here n=stair number, and the 4x4 stair case from the 8x8 grid can be substituted in to the formula staircase for the 8x8 grid.

Total for algebraic staircase= n+n+1+n+2+n+3+n+8+n+9+n+10+n+16+n+17+n+24 = 10n + 90

We can also evaluate that Stair number (n) = 5

By substitution stair total= 5+5+1+5+2+5+3+5+8+5+9+5+10+5+16+5+17+5+24= 10(5) + 90=140

Here is another example by using the algebraic stair case substitution:

As we can see here n=stair number, and the 4x4 stair case from the 9x9 grid can be substituted in to the formula staircase for the 9x9 grid.

Total for algebraic staircase= n+n+1+n+2+n+3+n+8+n+9+n+10+n+16+n+17+n+24 = 10n + 90

We can also evaluate that Stair number (n) = 37

By substitution stair total= 37+37+1+37+2+37+3+37+8+37+9+37+10+37+16+37+17+37+24= 10(37) + 90=460

Now that I have accomplished my investigation on the 8x8 grid with 4 step stair case, I am going to the 4 step stair investigation on the 7x7 grid.

By evaluating this stair case we can see that is the stair number is 1, then stair total = 1+2+3+4+8+9+10+15+16+22=90

Here we can see that the stair number=2

Then stair total= 2+3+4+5+9+10+11+16+17+23=100

Here we can see that the stair number = 3

Therefore stair total= 3+4+5+6+10+11+12+17+18+24= 110

The following table shows the stair total (T) depending on the relevant stair number for the 7x7 grid, with stair case number4. Which are from 1 to 5.

90    100    110     120      130

                       

                 +10                                  +10                               +10                           +10

Now that I solve the total, must include a multiple of 10. This will lead to the fact that the nth term has to have 10n in the formula, the extra is calculated by working out what is left over this will be +80.Therefore, the formula has to be T =10n + 80. By substituting the stair number to the nth term we get the stair total. The way how the formula was in algebraic terms is:

Formula to find out and nth term of any linear sequence = dn + (a-d) =T

a= First term in the linear sequence = 90

d= common difference = 10

n=stair number

T = stair total

By substitution, nth term was formed by: (10)(n) + (90-10) = 10n + 80

The way how the formula works is the following:

Which means that (10 x STAIR NUMBER) + 80 = STAIR TOTAL

Here is an alternative way to find the stair total of the 7x7 grid by using further algebraic method for the 4 step stair case.

As we can see here n=stair number, and the 4x4 stair case from the 7x7 grid can be substituted in to the formula staircase for the 7x7 grid.

Total for algebraic staircase= n+n+1+n+2+n+3+n+7+n+8+n+9+n+14+n+16+n+21 = 10n + 80

We can also evaluate that Stair number (n) = 22

By substitution stair total= 22+22+1+22+2+22+3+22+7+22+8+22+9+22+14+22+15+22+21= 10(22) + 80=300

Total for algebraic staircase= n+n+1+n+2+n+3+n+7+n+8+n+9+n+14+n+16+n+21 = 10n + 80

We can also evaluate that Stair number (n) = 25

By substitution stair total= 25+25+1+25+2+25+3+25+7+25+8+25+9+25+14+25+15+25+21= 10(25) + 80=330

Now I have done the investigation for the 4 step stair case for the 7x7, 8x8, 9x9, 10x10 grids. Now I am going to undertake my investigation for 2 step stair cases for the 77x, 8x8, 9x9, and 10x10 grids.

Firstly I will begin my investigation for the 2 step stair in the 10x10 grid:

Here we can see that stair number = 1

Therefore stair total= 1+2+11 = 14

Stair number = 2, therefore stair total= 2+3+12 = 17

Stair number= 3, therefore stair total = 13+3+4=20

The following table shows the stair total (T) depending on the relevant stair number for the 7x7 grid, with stair case number4. Which are from 1 to 5.

14        17      20        23      26

                       

                 +3                                  +3                               +3                           +3

Now that I solve the total, must include a multiple of 10. This will lead to the fact that the nth term has to have 3n in the formula, the extra is calculated by working out what is left over this will be +11.Therefore, the formula has to be T =3n + 11. By substituting the stair number to the nth term we get the stair total. The way how the formula was in algebraic terms is:

Formula to find out and nth term of any linear sequence = dn + (a-d) =T

a= First term in the linear sequence = 14

d= common difference = 3

n=stair number

T = stair total

By substitution, nth term was formed by: (3)(n) + (14-3) = 10n + 11

The way how the formula works is the following:

Which means that (3 x STAIR NUMBER) + 11 = STAIR TOTAL

Here is an alternative way to find the stair total of the 10x10 grid by using further algebraic method for the 2 step stair case.

As we can see here n=stair number, and the 2x2 stair case from the 10x10 grid can be substituted in to the formula staircase for the 10x10 grid.

Total for algebraic staircase= n+n+1+n+10 = 3n + 11

We can also evaluate that Stair number (n) = 9

By substitution stair total= 9+9+1+9+10= 3(9) +11= 38

Here is another example:

Total for algebraic staircase= n+n+1+n+10 = 3n + 11

We can also evaluate that Stair number (n) = 79

By substitution stair total= 79+79+1+79+10= 3(79) +11= 248

Now I am going to begin my investigation with the 9x9 grid size for the 2 step stair case:

Here we can see that the stair number = 1

Therefore the stair total = 1+2+10=13

Stair number = 2, therefore, stair total = 2+3+11= 16

Stair number = 3, therefore stair total = 3+4+12= 19

The following table shows the stair total (T) depending on the relevant stair number for the 7x7 grid, with stair case number4. Which are from 1 to 5.

13        16      19        22      25

                       

                 +3                                  +3                               +3                           +3

Now that I solve the total, must include a multiple of 10. This will lead to the fact that the nth term has to have 3n in the formula, the extra is calculated by working out what is left over this will be +10.Therefore, the formula has to be T =3n + 10. By substituting the stair number to the nth term we get the stair total. The way how the formula was in algebraic terms is:

Formula to find out and nth term of any linear sequence = dn + (a-d) =T

a= First term in the linear sequence = 13

d= common difference = 3

n=stair number

T = stair total

By substitution, nth term was formed by: (3)(n) + (13-3) = 10n + 11

The way how the formula works is the following:

Which means that (3 x STAIR NUMBER) + 10 = STAIR TOTAL

Here is an alternative way to find the stair total of the 9x9 grid by using further algebraic method for the 2 step stair case.

As we can see here n=stair number, and the 2x2 stair case from the 9x9 grid can be substituted in to the formula staircase for the 9x9 grid.

Total for algebraic staircase= n+n+1+n+9 = 3n + 10

We can also evaluate that Stair number (n) = 8

By substitution stair total= 8+8+1+8+9= 3(8) +10= 34

Here is another example:

Total for algebraic staircase= n+n+1+n+9 = 3n + 10

We can also evaluate that Stair number (n) = 71

By substitution stair total= 71+71+1+71+9= 3(71) +10= 223