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Math HL Type 1 Investigating Divisibility

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These are the portfolio questions

1. Factorize the expression P(n)=n^x-n for x = 2,3,4,5. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your result by showing whether P(k+1)-P(k) is always divisible by x. Using appropriate technology, explore more cases and make a conjecture for when n^x - n is divisible by x.

2. Explain how to obtain the entries in Pascal's triangle...State the relationship between the expression P(k+1)-P(k) and Pascal's traingle. Reconsider your conjecture.

Write an expression for the xth row of the Pascal's Triangle. You will have noticed that (x r) = k, k is a natural number. Determine when k is a multiple of x.

3. Make conclusions regarding the last result in part 2 and the form of proof by induction used in this assignment. Refine your conjecture if necessary, and prove it.

4. State the converse of your conjecture. Describe how you woul prove whether or not the converse holds.

i do not understand any of these questions can some help me please

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These are the portfolio questions

1. Factorize the expression P(n)=n^x-n for x = 2,3,4,5. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your result by showing whether P(k+1)-P(k) is always divisible by x. Using appropriate technology, explore more cases and make a conjecture for when n^x - n is divisible by x.

2. Explain how to obtain the entries in Pascal's triangle...State the relationship between the expression P(k+1)-P(k) and Pascal's traingle. Reconsider your conjecture.

Write an expression for the xth row of the Pascal's Triangle. You will have noticed that (x r) = k, k is a natural number. Determine when k is a multiple of x.

3. Make conclusions regarding the last result in part 2 and the form of proof by induction used in this assignment. Refine your conjecture if necessary, and prove it.

4. State the converse of your conjecture. Describe how you woul prove whether or not the converse holds.

i do not understand any of these questions can some help me please

ok so the first question you should just prove x= 2 3 5 by induction.

the second question, use nCr to show Pascal's triangle, and show that the entries are the coefficients of P(k + 1) - P(k)

Then state your conjecture which should be about x being prime etc.

the third question, dunno

4th, state the converse and show that its false for x=561

peace

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ok so the first question you should just prove x= 2 3 5 by induction.

the second question, use nCr to show Pascal's triangle, and show that the entries are the coefficients of P(k + 1) - P(k)

Then state your conjecture which should be about x being prime etc.

the third question, dunno

4th, state the converse and show that its false for x=561

peace

What is actually the relationship between the expression P(k+1)-P(k) and Pascal's Triangle?

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If you insert P(k+1)-P(k) into P(n) you will notice that you get the expansion of (k+1)^n without the first and the last term. So the coefficients of the elements will be the entries in Pascal's triangle.

Hope it helps. Good luck!

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Well, I don't know how much I'm allowed to help, but I did successfully finish this one earlier this year.

What monica said is exactly right. That's basically the form of induction that you need to use. Proving your hypothesis is the toughest part of this investigation (essentially proving that n^x-n is divisible by x when x is prime, or Fermat's Little Theorem), but look around and you'll be able to find how to do it.

Hint: look at the formula for nCr, and see how it is divisible by r when r is prime.

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what about the part that says "Using appropriate technology, explore more cases and make a conjecture for when n^x - n is divisible by x". what form of technology are we supposed to use?

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If you are that lost, ask your teacher for help, naynay. Unfortunately we will not solve your portfolio task for you. Also, don't revive old threads that are useless.

Closed.

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