G’day im currently working on a Math C assignment it consists of determining a formula for a rectangular Patten of tiles, then proving it by mathematical induction the formula is:
However no mater how hard I try I can’t work out how to prove it. Any help would be much appreciated
thanks Dan These are the Questions
By considering the tile patterns shown and other patterns in this design, determine a formula for the number of tiles in the nth pattern.
(the colour has nothing to do with the patten just shoes each square)
Prove the formula you obtained for the nth pattern.
The answer i got for Question one is 2n²-2n+1 i got it by dividing the patterns up in to equal proportions
then i worked out each section like a rectangle (n x (n-1))/2
multiplied that by 4 to account for each section then added 1 for the center one. it ended up like this:
then with a little rearrangement it became((n x (n-1))/2) x 4 + 1
its just Question 2 i don't understand how to do.2n²-2n+1
Here is a solution using a recursive function.
Define and for we get , where is the number of rectangles in each pattern.
To see how this works, consider what we may call the “main line”. The number of rectangles this “main line” is . We a to add 2 rectangles, one above and one below, each of the rectangles on the main line plus one on each end.
Thus : we start with the number we have add two for each on the main line plus two more, one on each end.
It can be shown that the closed form is .
Using the formula , look at picture 3:
Starting from picture 2, we add 2 rectangles above each square along the main line (including the middle one which already has two squares attached to it). Then we add 2 rectangles to the end to get the third picture.
Thanks Plato and tukeywilliams I think I got it does this sound right
Proving Via Mathematical Induction
2k² - 2k + 1
Explain how that Works Sn+1 = Sn + 2(2n – 1) +2
Then prove that 2k² - 2k + 1 Works when n = 1
2 x 1² - 2 x 1 + 1 = 1
Then Prove that it works when n = k + 1
2(k+1)² - 2(k+1) + 1 = 2k² - 2k + 1 + 4k2(k² + 2k + 2) –2k + 2 +1 = 2k² + 2k +12k² + 4k + 4 –2k + 3 = 2k² + 2k +12k² + 2k + 1 =2k² + 2k +1
Your second line followed incorrectly from your first line (i.e. ).