We’ll have a test on mathematical induction and the Binomial Theorem on Monday, February 8th.
To prepare for the test, complete p. 203–205 questions 20, 22, 24, 28, 26, 45, 47
We will discuss these questions in Thursday’s lesson.
Dr. McDonald's Mathematics Course Blog
Hello sir , will counting principles, including permutations and combinations be present on the exam this Monday
On this test, counting principles won’t be prominent. However, you will need to be familiar with the formula for \(_nC_r\).
No problem hopefully , thank you 🙂
Sir, I’m still unable to solve example 45 on page 196. Although I have progress, I’m still finding issues with the question. The solution in the text is not the most helpful either. I’m unable to figure out the link between the IH and the expansion for n=k+1
That proof is one of the more complicated examples. It relies on both the inductive hypothesis, along with the result that \[(\star) \begin{pmatrix}n\\r\end{pmatrix}+\begin{pmatrix}n\\r-1\end{pmatrix}=\begin{pmatrix}n+1\\r\end{pmatrix}\]
When the case for \(n=k+1\) is considered, the series is written out twice so that it can be added to itself (giving \(2\cdot2^k\) on the right, and allowing for the use of the result \(\star\) above, which gives, for example, \[\begin{pmatrix}k\\2\end{pmatrix}+\begin{pmatrix}k\\1\end{pmatrix}=\begin{pmatrix}k+1\\2\end{pmatrix}\]
Thus, on the right side we get \(2^{k+1}\), and we can rewrite all terms on the left so that they are of the form \[\begin{pmatrix}k+1\\i\end{pmatrix}\]
Hope that helps!