45 45+35+36=116
35 36
To do the nth term I have to times 3 by 35 then plus 11, which equals 116
I will now move on to a three-step stair.
82
72 73 82+73+72+64+63+62=416
62 63 64
81
71 72 81+72+71+63+62+61=410
61 62 63
80
70 71 80+71+70+62+61+60=404
60 61 62
I have found out that every time you move the three-step stair one to the left the product of the three-step stair goes down by six. This also means that every time you move the three-step stair to the right the product will go up by six.
33
23 24 33+24+23+15+14+13=122
13 14 15
43
33 34 43+33+34+23+24+25=182
23 24 25
53
43 44 33+34+35+43+44+53=242
33 34 35
The product of the last three three-step stairs tells me that when you move the stairs up by one the product of the three-step stair goes up by sixty. This also means that when you move the three-step stair down by one the product will go down by -60. I have also found out the nth term for the three-step stairs and it is 6n+44 I shall now test this.
78
68 69 58+59+60+68+69+78=392
58 59 60
To do the nth term I would have to times 6 by 58 then plus 44 which equals 392. Now I will move on to a four-step stair.
I shall now move on to a four-step stair
83
73 74
63 64 65 53+54+55+56+63+64+65+73+74+83=640
53 54 55 56
82
72 73
62 63 64 52+53+54+55+62+63+64+72+73+82=630
52 53 54 55
81
71 72
61 62 63 51+52+53+54+61+62+63+71+72+81=620
51 52 53 54
I have found out that every time you move the four-step stair one to the left the product of the four-step stair goes down by ten. This also means that every time you move the four-step stair to the right the product will go up by ten. I have also found out that all the products are even and I know that they will be even wherever you put a three or four-step stair it will always be even. On a three step stair wherever you put it on the grid there will always be 2 or 4 odd numbers and if you add 2 odds numbers together it makes a even and the same with 4 odd numbers and with a four-step stair there is ever 4 or 6 odd numbers.
43
33 34
23 24 25 13+14+15+16+23+24+25+33+34+43=240
13 14 15 16
53
43 44
33 34 35 23+24+25+26+33+34+35+43+44+53=340
23 24 25 26
63
53 54
43 44 45 33+34+35+36+43+44+45+53+54+63=440
33 34 35 36
I have found out that the product of the four-step stairs tells me that when you move the stairs up by one the product of the four-step stair goes up by one hundred. This also means that when you move the four-step stair down by one the product will go down by -100. I have also found the nth term for the four-step stair is 10n+110. I shall now test the nth term.
56
46 47
36 37 38 26+27+28+29+36+37+38+46+47+56=370
26 27 28 29
To do the nth term I have to times 10 by 26 then plus 110 which equals 370.
So I will now move on to a five-step stair.
83
73 74
63 64 65 43+44+45+46+47+53+54+55+56+63+64+65+73+74+83=865
53 54 55 56
43 44 45 46 47
82
72 73
62 63 64 42+43+44+45+46+52+53+54+55+62+63+64+72+73+82=850
52 53 54 55
42 43 44 45 46
81
71 72
61 62 63
51 52 53 54 41+42+43+44+45+51+52+53+54+61+62+63+71+72+81=835
41 42 43 44 45
I have found out that every time you move the five-step stair one to the left the product of the three-step stair goes down by fifteen. This also means that every time you move the three-step stair to the right the product will go up by fifteen, but the product is now not always an even number it is sometimes an even or an odd number. This is because there is not always an even amount of odd numbers, there is some times an odd amount of odd numbers. I predict that when you move a five-step stair up by one the product will go up by 150. This is because when you move it up each number is going up by 10 and there is 15 numbers in the number so this means that it would go up by 150.
53
43 44
33 34 35 53+43+44+33+34+35+23+24+25+26+13+14+15+16+17=415
23 24 25 26
13 14 15 16 17
63
53 54
43 44 45 63+53+54+43+44+45+33+34+35+36+23+24+25+26+27=565
33 34 35 36
23 24 25 26 27
73
63 64
53 54 55 73+63+64+53+54+55+43+44+45+46+33+34+35+36+37=715
43 44 45 46
33 34 35 36 37
I have found out that every time you move the 5-step stair up by 1 the product goes up by 150, this means that when you move the 5-step stair down by one the product goes down by 150. I have also found out the nth term for this which is 15n+220, I shall now test this.
64
54 55
44 45 46 64+54+55+44+45+46+34+35+36+37+24+25+26+27+28=580
34 35 36 37
24 25 26 27 28
To do the nth term I have to times 15 by 24 then plus 220 which equals 580. The nth term is correct.
With this information I can find out any nth term. I shall now try and find out more nth terms.
Conclusion
Overall I have found out the nth terms for a 1 to a ten-step stair and some patterns with the nth term. I have also found out how much a 1 to a 5-step stair product changes if you move it up, down, left and right.
