Induction

3.1 Prove, using induction but also without it, that $n(n+1)$ is an even number for every non-negative integer $n$.

Without induction:

$n$ and $n+1$ are guaranteed to be one even and one odd number. If one factor is even, the product is even.

With induction:

$(n+1)\ast (n+2)$ is $n^2+3n+2$. Subtract from $n(n+1)$ which is $n^2+n$ gets us $n^2+2$ or $n(n+1)$. Thus, our next value must be even as well.

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3.2 Prove by induction that the sum of the first $n$ positive integers is $n(n+1)/2$.

For the base case of 1: $P(1)=1$. For any positive integer $k$, try to prove that $\mathrm{1..}k$ + $k+1$ = $(k+1)(k+2)/2$.

We know that $P(k)$ = the correct sum, so replace equivalence: $k(k+1)/2+(k+1)=(k+1)(k+2)/2$.

Simplifying the left:

$k(k+1)/2+2(k+1)/2$ $(k(k+1)+2(k+1)/2$ $(k+1)(k+2)/2$

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3.3 Observe that the number $n(n+1)/2$ is the number of handshakes among $n+1$ people. Suppose that everyone counts only handshakes with people older than him/her (pretty snobbish, isn’t it?). Who will count the largest number of handshakes? How many people count 6 handshakes? Give a proof of the result of exercise 3.1, based on your answer to these questions.

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3.4 Give a proof of exercise 3.1, based on figure 3.

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3.5 Prove the following identity: $1\cdot 2+2\cdot 3+3\cdot 4+\cdots +(n-1)\cdot n=\frac{(n-1)\cdot n\cdot (n+1)}{3}$

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3.6 Use the method of the little Gauss to give a third proof of the formula in exercise 3.1.

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3.7 How would the little Gauss prove the formula for the sum of the first $n$ odd numbers $(2)$?

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3.8 Prove that the sum of the first $n$ squares $(1+4+9+\cdots +n^2)$ is $n(n+1)(2n+1)/6$.

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3.9 Prove that the sum of the first $n$ powers of 2 (starting with $1=2^0$) is $2^n-1$.

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