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 have6nin the formula, the extra is calculated by working out what is left over this will be+40.Therefore, the formula has to be T =6n + 40. By substituting the stair number to the nth term we get the stair total. Here we can see that is clearly evident that the nth term for the 9×9 grid has decreased by 4 as compared to the 10×10 grid. The way how the formula work is the following:Which means that (6 x STAIR NUMBER) + 40 = STAIR TOTAL.Now that I have worked out the formula for the 9×9 grid I am going to use the formula in random staircases with random stair numbers. And those are:Stair number = 61Stair total = (6×61) + 40 = 406, or alternatively, 61+62+63+70+71+79= 406Stair number=7Stair total= (6×7) + 40= 82With out the nth term the stair total= 7+8+9+16+17+25=82Stair number=55With out the nth term stair total= 55+56+57+64+65+73= 370With nth term, stair total= (6×55) +40=370Here is an alternative way to find the stair total of the 9×9 grid by using further algebraic method.As we can see here n=stair number, and the 3×3 stair case from the 9×9 grid can be substituted in to the formula staircase for the 9×9 grid.Total for algebraic staircase= n+n+1+n+2+n+9+n+10+n+18= 6n+40We can also evaluate that Stair number (n) = 55By substitution stair total= 55+55+1+55+2+55+9+55+10+55+18=370Here is another example:By substitution, stair total= 7+7+1+7+2+7+9+7+10+7+18=82Here is another example:It is apparent that the stair number = 39By substitution stair total= 39+39+1+39+2+39+9+39+10+39+18=274Now that I have done my further investigation for the 3 step stair for the 9×9 grid, now I am going to undertake further investigation for 3 step stair for the 8×8 grid.Here is an 8×8 grid showing the stair total and the stair number:Stair number (n)=1By 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 8×8 grid with the stair number 1.Stair number= 2Whereas the stair total= 2+3+4+10+11+18 = 48 = TStair number=3Whereas the stair total= 3+4+5+11+12+19= 54 = TThe following table shows the stair total (T) depending on the relevant stair number for the 8×8 grid. Which are from 1 to 5.42 48 54 60 66+6 +6 +6 +6It 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 8×8 grid has decreased by 4 as compared to the 9×9 grid. The way how this formula works is the following:Which means that (6 x STAIR NUMBER) + 40 = STAIR TOTALNow that I have worked out the formula for the 8×8 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= 46Whereas stair total= 46+47+48+54+55+62=312But by using the nth term, stair total= (6×46) + 36= 312Stair number=41Whereas stair total= 41+42+43+49+50+57= 282But by using the nth term, stair total= (6×41) + 36= 282Stair number=27Whereas stair total= 27+28+29+35+36+43=198But by using the nth term, stair total= (6×27) + 36= 198Here is an alternative way to find the stair total of the 8×8 grid by using further algebraic method.As we can see here n=stair number, and the 3×3 stair case from the 8×8 grid can be substituted in to the formula staircase for the 8×8 grid.Total for algebraic staircase= n+n+1+n+2+n+8+n+9+n+16= 6n+36We can also evaluate that Stair number (n) = 6By substitution stair total= 6+6+1+6+2+6+8+6+9+6+16=72Here is another example:Here we can see that n= 46By substitution, stair total= 46+46+1+46+2+46+8+46+9+46+16= 312Now that I have done my further investigation for the 3 step stair for the 8×8 grid, now I am going to undertake further investigation for 3 step stair for the 7×7 grid.1589123Stair number=1Whereas stair total= 1+2+3+8+9+15= 38Here we can see that stair number=2Therefore, stair total= 2+3+4+9+10+16=44Here we can see that stair number=3Therefore, stair total= 3+4+5+10+11+17= 50Stair number (n) = 4Therefore, stair total= 4+5+6+11+12+18 = 56The following table shows the stair total (T) depending on the relevant stair number for the 7×7 grid. Which are from 1 to 5.38 44 50 56 62+6 +6 +6 +6It 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 7×7 grid has decreased by 4 as compared to the 8×8 grid. The way how this formula works is the following:Which means that (6 x STAIR NUMBER) + 32 = STAIR TOTALNow that I have worked out my nth term for the 3 step stair case in the 7×7 grid size, I am going to use this equation into random 3 step stair case in the 7×7 grid, in order to find out the stair total and to see if the formula works.Stair number=33Stair total= 33+34+35+40+41+47= 230But by using the nth term our stair total= (6×33) + 32=230Stair number = 29Stair total= 29+30+31+36+37+43 = 206By using nth term our stair total= (6×29) +32= 206Stair number = 19Stair total= 19+20+21+26+27+33= 146By using our formula the stair total= (6×19) + 32= 146Here is an alternative way to find the stair total of the 7×7 grid by using further algebraic method.As we can see here n=stair number, and the 3×3 stair case from the 7×7 grid can be substituted in to the formula staircase for the 7×7 grid.Total for algebraic staircase= n+n+1+n+2+n+7+n+8+n+14= 6n+32We can also evaluate that Stair number (n) = 16By substitution stair total= 16+16+1+16+2+16+7+16+8+16+14= 128Here is another example:n+14n+7n+8nn+1n+2Here we can see that n= 12By substitution, stair total= 12+12+1+12+2+12+7+12+8+12+14= 104Now I have done the investigation for the 3 step stair case for the 7×7, 8×8, 9×9, 10×10 grids. Now I am going to undertake my investigation for 4 step stair cases for the 77x, 8×8, 9×9, and 10×10 grids.I will once again use (T) for my stair total, and (N) for my stair number.I am going to start with the 10×10 grid.N= 1The 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=2Stair total= 2+3+4+5+12+13+14+22+23+32= 130Stair number=3Stair total= 3+4+5+6+13+14+15+23+24+33= 140Stair number= 4Stair total = 4+5+6+7+14+15+16+24+25+34= 150The following table shows the stair total (T) depending on the relevant stair number for the 10×10 grid, with stair case number4. Which are from 1 to 5.120 130 140 150 160+10 +10 +10 +10It 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 TOTALNow that I have worked out my nth term for the 4 step stair case in the 10×10 grid size, I am going to use this equation into random 4 step stair case in the 10×10 grid, in order to find out the stair total and to see if the formula works.Stair number=57Stair total = 57+58+59+60+67+68+69+77+78+87 = 680But by using the nth term, stair total= (10×57) + 110 = 680Stair number = 51Stair total = 51+52+53+54+61+62+63+71+72+81 = 620By using nth term stair total = (10×51) + 110 = 620Stair number = 7Stair total = 7+8+9+10+17+18+19+27+28+37 = 180By using the nth term stair total = (10×7) + 110 = 180Here is an alternative way to find the stair total of the 10×10 grid by using further algebraic method for the 4 step stair case.As we can see here n=stair number, and the 4×4 stair case from the 10×10 grid can be substituted in to the formula staircase for the 10×10 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 + 110We can also evaluate that Stair number (n) = 36By substitution stair total= 36+36+1+36+2+36+3+36+10+36+11+36+12+36+20+36+21+36+30= 470Here is another example:n+30n+20n+21n+10n+11n+12Nn+1n+2n+3As we can see here n=stair number, and the 4×4 stair case from the 10×10 grid can be substituted in to the formula staircase for the 10×10 grid.We can also evaluate that Stair number (n) = 17By substitution stair total= 17+17+1+17+2+17+3+17+10+17+11+17+12+17+20+17+21+17+30= 280Now that I have accomplished my investigation on the 10×10 grid with 4 step stair case, I am going to the 4 step stair investigation on the 9×9 grid.2819201011121234Stair number (N) =1, stair total=1+2+3+4+10+11+12+19+20+28= 110Stair number (n) = 2Stair Total = 2+3+4+5+11+12+13+20+21+29= 120.Stair number = 3Stair total = 3+4+5+6+12+13+14+21+22+30= 130Stair number=4Stair total= 4+5+6+7+13+14+15+22+23+31=140The following table shows the stair total (T) depending on the relevant stair number for the 9×9 grid, with stair case number4. Which are from 1 to 5.110 120 130 140 150+10 +10 +10 +10It 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) =Ta= First term in the linear sequence = 110d= common difference = 10n=stair numberT = stair totalBy substitution, nth term was formed by: (10)(n) + (110-10) = 10n + 100Now that I have figured out the formula for the 4 step stair for the 9×9 grid, I am going to test this formula in to random stair cases in the 9×9 grid.Stair total = 51+52+53+54+60+61+62+69+70+18= 610By formula, stair total = (10×51) + 100 = 610Stair total = 46+47+48+49+55+56+57+64+65+73 = 560By using nth term, stair total = (10×46) + 100= 560Here is an alternative way to find the stair total of the 9×9 grid by using further algebraic method for the 4 step stair case.51424333343524252627As we can see here n=stair number, and the 4×4 stair case from the 9×9 grid can be substituted in to the formula staircase for the 9×9 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 + 100We can also evaluate that Stair number (n) = 24By substitution stair total= 24+24+1+24+2+24+3+24+9+24+10+24+11+24+18+24+19+24+27= 340 = TNow that I have accomplished my investigation on the 9×9 grid with 4 step stair case, I am going to the 4 step stair investigation on the 8×8 grid.Here we can see that, stair number=1Stair total= 1+2+3+4+9+10+11+17+18+25= 1002719201112133456Here we can see that stair number = 2 Stair number = 3Whereas, 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= 110The following table shows the stair total (T) depending on the relevant stair number for the 8×8 grid, with stair case number4. Which are from 1 to 5.100 110 120 130 140+10 +10 +10 +10Now 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) =Ta= First term in the linear sequence = 100d= common difference = 10n=stair numberT = stair totalBy substitution, nth term was formed by: (10)(n) + (100-10) = 10n + 90The way how the formula works is the following:Which means that (10 x STAIR NUMBER) + 90 = STAIR TOTALHere is an alternative way to find the stair total of the 8×8 grid by using further algebraic method for the 4 step stair case.As we can see here n=stair number, and the 4×4 stair case from the 8×8 grid can be substituted in to the formula staircase for the 8×8 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 + 90We can also evaluate that Stair number (n) = 5By 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=140Here is another example by using the algebraic stair case substitution:As we can see here n=stair number, and the 4×4 stair case from the 9×9 grid can be substituted in to the formula staircase for the 9×9 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 + 90We can also evaluate that Stair number (n) = 37By 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=460Now that I have accomplished my investigation on the 8×8 grid with 4 step stair case, I am going to the 4 step stair investigation on the 7×7 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=90Here we can see that the stair number=2Then stair total= 2+3+4+5+9+10+11+16+17+23=100Here we can see that the stair number = 3Therefore stair total= 3+4+5+6+10+11+12+17+18+24= 110The following table shows the stair total (T) depending on the relevant stair number for the 7×7 grid, with stair case number4. Which are from 1 to 5.90 100 110 120 130+10 +10 +10 +10Now 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) =Ta= First term in the linear sequence = 90d= common difference = 10n=stair numberT = stair totalBy substitution, nth term was formed by: (10)(n) + (90-10) = 10n + 80The way how the formula works is the following:Which means that (10 x STAIR NUMBER) + 80 = STAIR TOTALHere is an alternative way to find the stair total of the 7×7 grid by using further algebraic method for the 4 step stair case.43363729303122232425As we can see here n=stair number, and the 4×4 stair case from the 7×7 grid can be substituted in to the formula staircase for the 7×7 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 + 80We can also evaluate that Stair number (n) = 22By 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=30046394032333425262728Total for algebraic staircase= n+n+1+n+2+n+3+n+7+n+8+n+9+n+14+n+16+n+21 = 10n + 80We can also evaluate that Stair number (n) = 25By 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=330Now I have done the investigation for the 4 step stair case for the 7×7, 8×8, 9×9, 10×10 grids. Now I am going to undertake my investigation for 2 step stair cases for the 77x, 8×8, 9×9, and 10×10 grids.Firstly I will begin my investigation for the 2 step stair in the 10×10 grid:Here we can see that stair number = 1Therefore stair total= 1+2+11 = 14Stair number = 2, therefore stair total= 2+3+12 = 17Stair number= 3, therefore stair total = 13+3+4=20The following table shows the stair total (T) depending on the relevant stair number for the 7×7 grid, with stair case number4. Which are from 1 to 5.14 17 20 23 26+3 +3 +3 +3Now 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) =Ta= First term in the linear sequence = 14d= common difference = 3n=stair numberT = stair totalBy substitution, nth term was formed by: (3)(n) + (14-3) = 10n + 11The way how the formula works is the following:Which means that (3 x STAIR NUMBER) + 11 = STAIR TOTALHere is an alternative way to find the stair total of the 10×10 grid by using further algebraic method for the 2 step stair case.n+10nn+1As we can see here n=stair number, and the 2×2 stair case from the 10×10 grid can be substituted in to the formula staircase for the 10×10 grid.Total for algebraic staircase= n+n+1+n+10 = 3n + 11We can also evaluate that Stair number (n) = 9By substitution stair total= 9+9+1+9+10= 3(9) +11= 38Here is another example:Total for algebraic staircase= n+n+1+n+10 = 3n + 11We can also evaluate that Stair number (n) = 79By substitution stair total= 79+79+1+79+10= 3(79) +11= 248Now I am going to begin my investigation with the 9×9 grid size for the 2 step stair case:Here we can see that the stair number = 1Therefore the stair total = 1+2+10=13Stair number = 2, therefore, stair total = 2+3+11= 16Stair number = 3, therefore stair total = 3+4+12= 19The following table shows the stair total (T) depending on the relevant stair number for the 7×7 grid, with stair case number4. Which are from 1 to 5.13 16 19 22 25+3 +3 +3 +3Now 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) =Ta= First term in the linear sequence = 13d= common difference = 3n=stair numberT = stair totalBy substitution, nth term was formed by: (3)(n) + (13-3) = 10n + 11The way how the formula works is the following:Which means that (3 x STAIR NUMBER) + 10 = STAIR TOTALHere is an alternative way to find the stair total of the 9×9 grid by using further algebraic method for the 2 step stair case.As we can see here n=stair number, and the 2×2 stair case from the 9×9 grid can be substituted in to the formula staircase for the 9×9 grid.Total for algebraic staircase= n+n+1+n+9 = 3n + 10We can also evaluate that Stair number (n) = 8By substitution stair total= 8+8+1+8+9= 3(8) +10= 34Here is another example:Total for algebraic staircase= n+n+1+n+9 = 3n + 10We can also evaluate that Stair number (n) = 71By substitution stair total= 71+71+1+71+9= 3(71) +10= 223