Could anyone please help me on the following problem:
- 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 results by showing whether $P(k+1)-P(k)$ is always divisible by $x$. Using appropriate technology, explore more cases, summarize your results and make a conjecture for when $n^x-n $is divisible by$ x$. (Not divisible by$\ 4$. Thus I concluded that $x$ is prime.)
2.Explain how to obtain the entries in Pascal's Triangle, and using appropriate technology, generate the first 15 rows. State the relationship between the expression $P(k+1)-P(k)$ and Pascal's Triangle. Reconsider your conjecture and revise if necessary. (Here, my previous conjecture appeared incorrect, because $x$=prime numbers did not work out for 11 and 13 from the Triangle).
And on this part I am experiencing a bit of trouble as I am saying that $x=2,3,5,7$. But this does not work for all $r$...I decided that $k$ is a multiple of $x$ when $x=2, 3, 5, 7, 9$. However, $r$ is different for each of the $x$ ...
Write an expression for the xth row of Pascal's Triangle. You will have noticed that ($x$ choose $r)=k$, $k$ is $N$. Determine when $k$ is a multiple of $x$.
- make conclusions regarding the last result in part 2 and the form of proof by induction used i nthis assignment. Refine your conjecture if necessary, and prove it.
I believe that I am not getting this correctly, help is greatly appreciated!
Thanks in advance for your help.