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I have 3 math induction proofs I have been struggling with for a while. I understand how to do summation proofs but these ones, I can't find a general pattern to solve. Please help.

1) $D(n) = {n(n-3) \over2}$ for all $n \ge 3$ This is for the n diagonals of a polygon.

2)$\binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \ldots + \binom{n}{n} = 2^n$ for all $n\ge 0$

3)$n^2 + 1 \ge 3n$ for all $n \ge 3$; For this one, I got stuck when I did this:

$$\begin{align} k^2 + 1 &\ge 3k \\ (k+1) ^ 2 +1 &\ge 3(k+1) \\ (k+1) ^ 2 &\ge k^2 \ge 3k \\ (k+1) ^ 2 &\ge k^2 + 3 \ge 3k + 3 \,\, \text{(? I don't know how to proceed from here ?)} \end{align}$$

Please give me possible solutions and techniques to solve these kind of problems.

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  • $\begingroup$ It would be best to split this up into 3 separate questions. Also, in 1, what is $D(n)$? $\endgroup$ – Calvin Lin Oct 9 '13 at 0:41
  • $\begingroup$ I have updated your post to LaTeX, please see that the updates are correct. $\endgroup$ – Jeel Shah Oct 9 '13 at 1:03
  • $\begingroup$ @gekkostate Why the brackets in the numerator of $D(n)$? $\endgroup$ – Vedran Šego Oct 9 '13 at 1:12
  • $\begingroup$ @VedranŠego That's how it was originally and I did not want to make any changes so I simply surrounded it with \$\$ and left it alone. $\endgroup$ – Jeel Shah Oct 9 '13 at 1:16
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    $\begingroup$ @gekkostate Ah, sorry. I didn't notice the brackets before the edit. $\endgroup$ – Vedran Šego Oct 9 '13 at 1:18
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I assume that $D(n)$ is the number of diagonals in an $n$-sided convex polygon. Descriptively: each vertex of the polygon can be connected to $n-3$ other (connecting it with itself or its neighbors will not produce diagonals). That way, you make each diagonal twice, hence the formula.

If you want to use induction, then check for $n=3$, and then assume it works for all $k$-sided convex polygons where $k < n$. Look at the $n$-sided one.

We can pick any $n-1$ vertices, and the polygon they span has $(n-1)((n-1)-3)/2$ diagonals (by the assumption). But, one of its sides will also become a diagonal of the bigger polygon. Now, we're just missing $n-3$ diagonals from the $n$-th vertex, so in total we have:

\begin{align*} \frac{(n-1)((n-1)-3)}{2} + 1 + (n-3) &= \frac{(n-1)(n-4) + 2 + 2(n-3)}{2} \\ &= \frac{n^2 - 5n + 4 + 2 + 2n - 6}{2} = \frac{n^2 - 3n}{2} \\ &= \frac{n(n-3)}{2}. \end{align*}

If you want the second one using induction, use the recursive formula for binomial coefficients:

$$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}.$$

Edit: Since you asked how to do this (in comment to nsanger), I'm expanding that part of the answer. The idea is to note that the two terms on the right hand side are the same, only with a shift in $k$. In other words:

$$\sum_{k=1}^n \binom{n-1}{k-1} = \sum_{k=\color{red}{0}}^{\color{red}{n-1}} \binom{n-1}{\color{red}{k}}.$$

Now, we just need to play with the edge cases to get the sums to have the same limits: \begin{align*} \sum_{k=0}^n \binom{n}{k} &= \binom{n}{0} + \binom{n}{n} + \sum_{k=\color{red}{1}}^{\color{red}{n-1}} \binom{n}{k} = 1 + 1 + \sum_{k=1}^{n-1} \binom{n-1}{k} + \sum_{k=1}^{n-1} \binom{n-1}{k-1} \\ &= \binom{n-1}{0} + \sum_{k=1}^{n-1} \binom{n-1}{k} + \binom{n-1}{n-1} + \sum_{k=1}^{n-1} \binom{n-1}{k-1} \\ &= \sum_{k=\color{red}{0}}^{n-1} \binom{n-1}{k} + \sum_{k=1}^{\color{red}{n}} \binom{n-1}{k-1} \\ &= \Big\{ \text{Use the above formula for the second sum} \Big\} \\ &= \sum_{k=0}^{n-1} \binom{n-1}{k} + \sum_{k=0}^{n-1} \binom{n-1}{k} = 2\sum_{k=0}^{n-1} \binom{n-1}{k} \\ &= \Big\{ \text{Use the induction hypothesis} \Big\} \\ &= 2 \cdot 2^{n-1} = 2^n. \end{align*}

The third one was answered by nsanger, so I won't be repeating that.

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I don't know what the function $D(n)$ is, so I can't help you on the first problem.

To prove the second equality you don't actually need induction, you can just use the binomial theorem, which states that $$(a+b)^n=\sum_{k=0}^n \binom{n}{k}a^{n-k}b^k$$ If you plug in $a = b = 1$ then you are left with $$2^n = (1+1)^n=\sum_{k=0}^n\binom{n}{k}1^{n-k}1^{k}=\sum_{k=0}^n\binom{n}{k}=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\cdots+\binom{n}{n}$$

To prove the third one by induction, first note that in the case $n = 3$ you have $n^2 + 1=10>9=3n$, so the theorem holds for the base case. To prove it for the $n+1$ case, since you are given that $n^2 + 1 > 3n$, and that $n\geq 3$, you can say that $2n +1> 3$ so therefore $$n^2+1>3n\implies (n^2+1)+(2n+1)>3n+3$$ $$(n^2+2n+1)+1>3(n+1)$$ $$(n+1)^2+1>3(n+1)$$ which shows the theorem holds for the $n+1$ case as well, concluding the proof.

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  • $\begingroup$ Actually I need to prove the second one using induction and pascals identity. Any idea on how to do that ? $\endgroup$ – oneCoderToRuleThemAll Oct 9 '13 at 1:27
  • $\begingroup$ Okay, then try using the identity mention by Vedran. You might find it more useful in the form $$\binom{n+1}{k}=\binom{n}{k-1} + \binom{n}{k}.$$ Then the problem is reduced to evaluating the sum $$\sum_{k=0}^{n+1} \binom{n+1}{k}=\sum_{k=0}^{n+1} \binom{n}{k} + \sum_{k=0}^{n+1} \binom{n}{k-1}$$ Then try using your inductive hypothesis and some algebra. $\endgroup$ – user71641 Oct 9 '13 at 1:43
  • $\begingroup$ @GdgamesGamers Please see the edit of my answer. $\endgroup$ – Vedran Šego Oct 9 '13 at 9:52
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1) The base case is a triangle, having no diagonal. Then take a polygon of $n$ vertices, add a vertex on an edge and displace it. This makes $n-1$ new diagonals appear ($n-2$ to the new vertex plus the original edge). This matches $D(n+1)-D(n)=n-1$.

enter image description here

2) The base case $n=0$ is trivial, $2^0=1$. Now take a row in Pascal's triangle. You know that it is obtained by adding pairwise the elements of the previous row. So the sum of the row is twice that of the previous row.

enter image description here

3) The base case is true, as $10\ge9$. Then subtract memberwise the inequalities for $n+1$ and $n$ (don't forget to swap the sides): $$\begin{align} &(n+1)^2+1&\ge&3(n+1)\\ -\ \ \ \ \ \ &n^2+1&\ge&3n\\ =\ \ \ \ \ \ &n^2-n+2&\ge&-n^2+3n\\ \end{align}$$

The last relation can be rewritten $2(n-1)^2\ge0$, which is always true.

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