Wednesday, January 13, 2021

Number stairs

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Maths Coursewor.Maths Coursewor.Part .Finding the formula (1. Help with essay on number stairs


Finding the formula for step stairs on a 10x10 number grid using graphical method.n+.n+10n+1.nn+1n.This is a step stair in algebraic form. The letter n is the stair position. This shows that the bottom left box of the stairs is the stair position. The next box to the right is 1 more than the stair position, the next box to the right of that is more, etc.From this we can sa.t = stair tota. = 6n + 4.Exampl.4.5.56.This is the step stair, where the step position is 1.Using addition we can work out the stair tota.t = stair tota. = 5 + 6 + 7 + 5 + 6 + 4. = 1.This can also be worked out using the formula abov.t = stair tota. = 6n + 4. = 6 x 5 + 4. = 1.Finding the formula (.Finding the formula for step stairs on a 10x10 number grid using sequence..11..


1 + + + 11 + 1 + 1=5.


.1.


+ + 4 + 1 + 1 + =5.


.11.


+ 4 + 5 + 1 + 14 + =..141.45.


4 + 5 + 6 + 14 + 15 + 4=6.To find the formula using sequences, we can put consecutive stair totals in sequence, and try to solve the.505666.


66 6᠖this shows the formula must contain 6.By writing out t = 6n and finding the difference between it and the sequence above, we can find out the formula.50 -6=4.56 -1=4.6 -18=4.68 -4=4.This numbers on the left are from the sequence above and the numbers in the middle column are from the sequence t = 6n. The fact that they all make 44, means that + 44 must be in the equation too.So with the 6n from the first sequence, and the + 44 from the second stage of working out, the final formula works out t.t = stair tota. = 6n + 4.Part.Finding the formul.w = width of the number gri.1 step stai. t = . step stai.n+.nn+. t = n + . step stai.n+.n+wn+w+.nn+1n.


t = 6n + 4w + .4 step stai.n+w.n+wn+w+.n+wn+w+1n+w.nn+1n+n.


t = 10n + 10w + 1.5 step stai.n+4w n+wn+w+1n+wn+w+1n+w+.n+wn+w+1n+w+n+w.nn+1n+n+n+.


t = 15n + 0w + .6 step stai.n+5.n+4wn+4w+1 n+wn+w+1n+w+ n+wn+w+1n+w+n+w+4n+wn+w+1n+w+n+w+n+w+4.nn+1n+n+n+4n+.t = 1n + 5w + .As you can see, the numbers in the formulas change as the step height changes, so we need to find an overall formula. First we need to break it dow.1 stept = . stept = n + w + . stept = 6n + 4w + .4 stept = 10n + 10w + 1.5 stept = 15n + 0w + .6 stept = 1n + 5w + .To work out the first section of the formula, we need to put them in sequenc.161015.5 6.


1111᠖ Formula must contain 1/n² or 0.5n.If we write out t = 1/n² and find the differences between it and the above sequence, we can find out the rest of the formul.1-0.5=0..


-=.6-4.5=1..10-8.15-1.5=..1-18.This tells us that the difference between the sequence above and 0.5n² is 0.5n, so the final formula looks like thi.t = 0.5n² + 0.5.Next, we need to work out the second part of the formul.1 stept = . stept = n + w + . stept = 6n + 4w + .4 stept = 10n + 10w + 1.5 stept = 15n + 0w + .6 stept = 1n + 5w + .We need to put these in sequence to work them out to.014100.


1610 1.4 . 11 1᠖ Formula must contain 1/6n.Next we write out 1/6n³ and find the differences between it and the above sequenc.0-1/6=1/.1-1 ͓=/.4-4 ½=/.10-10 U=4/.0-0 5/6=5/.5-6=.This tells us that the difference between 1/6n³ and the formula above is 1/6n, so the final equation works out like thi.t = n(0.5h² + 0.5h) + w(1/6h³ - 1/6h) + (1/6h³ - 1/6h.Exampl.4.5.56.t = step tota.n = step position (bottom left box.w = number grid widt.h = step heigh.t = n(0.5h² + 0.5h) + w(1/6h³ - 1/6h) + (1/6h³ - 1/6h. = 5(0.5() + 0.5()) + 10(1/6(7) 1/6()) + (1/6(7) 1/6(). = 150 + 40 + . = 1.The En..


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