1. ## Mathematical Induction

Mathematical Induction

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:

2n²-2n+1

However no mater how hard I try I can’t work out how to prove it. Any help would be much appreciated

thanks Quill

2. Originally Posted by Quill
Mathematical Induction

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:

2n²-2n+1

However no mater how hard I try I can’t work out how to prove it. Any help would be much appreciated

thanks Quill
This is going to be awfully hard to help you with unless you explain what that formula is supposed to refer to...

-Dan

3. thanks Dan These are the Questions

Question 1
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)

Question 2
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:
((n x (n-1))/2) x 4 + 1
then with a little rearrangement it became
2n²-2n+1
its just Question 2 i don't understand how to do.

4. Here is a solution using a recursive function.
Define $S_1 = 1$ and for $n \ge 1$ we get $S_{n+1} = S_{n} + 4n$, where $S_n$ 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 $1,3,5,\cdots,2n-1$. 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 $S_{n+1} = S_{n} + 2(2n-1) +2$: 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 $2n^2 - 2n+1$.

5. Okay so the “main line” is the bit through the center

I don’t quite understand what you mean by

“We a to add 2 rectangles, one above and one below, each of the rectangles on the main line plus one on each end.”

Are you referring to these bits

6. Using the formula $S_{n+1} = S_n + 2(2n-1) + 2$, 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.

7. 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 + 4k
2(k² + 2k + 2) –2k + 2 +1 = 2k² + 2k +1
2k² + 4k + 4 –2k + 3 = 2k² + 2k +1
2k² + 2k + 1 =2k² + 2k +1

8. yes that is correct.

9. Originally Posted by Quill
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 + 4k
2(k² + 2k + 2) –2k + 2 +1 = 2k² + 2k +1
2k² + 4k + 4 –2k + 3 = 2k² + 2k +1
2k² + 2k + 1 =2k² + 2k +1
Proof: We use induction on $n$. Base case: For $n = 1$, $2(1)^2 -2(1)+1 = 1$. Inductive step: Suppose now as inductive hypothesis that $2k^2 -2k+1$ is true for some positive integer $k$. Then $2(k+1)^2 - 2(k+1) + 1 = 2k^2 + 4k + 2 - 2k - 2 + 1 = 2k^2 +2k + 1$ as required. Conclusion: Hence, by induction, $2n^2 +2n + 1$ is true for all positive integers $n$. $\square$

Your second line followed incorrectly from your first line (i.e. $2(k^2 + 2k +2) - 2k +2 +1 = 2k^2 + 2k +1$).

10. Cool thank you very much

*Edit* Just wondering how would I reference this sight in a assignment and would it be considered plagiarism because u showed me how to do the question.