Explain how you know that (n+1)^k -n^k has degree k-1
Notice that expanded the first term is
n^k + 2n(k-1) and so on with diminishing powers of n.
The second term is n^k
so, subtracting the second from the first, the n^k term goes away, leaving the next highest power of n to be k-1
To understand why (n+1)^k - n^k has degree k-1, let's break it down step by step.
Starting with (n+1)^k, we can expand it using the binomial theorem:
(n+1)^k = n^k + (k choose 1) * n^(k-1) * 1 + (k choose 2) * n^(k-2) * 1^2 + ... + (k choose k-1) * n * 1^(k-1) + 1^k
When we subtract n^k from (n+1)^k, the n^k term cancels out, leaving us with:
(n+1)^k - n^k = (k choose 1) * n^(k-1) * 1 + (k choose 2) * n^(k-2) * 1^2 + ... + (k choose k-1) * n * 1^(k-1) + 1^k
Since each term in this expression has a power of n that is less than or equal to k-1, the highest power of n in the entire expression is k-1.
Therefore, we conclude that (n+1)^k - n^k has degree k-1.