Draw how to tile the whole plane with figures, composed from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), and \(5\times 5\) where squares of all sizes are used the same amount of times in the design of the figure.
Show that if \(1+2+\dots+n = \frac{n(n+1)}{2}\), then \(1+2+\dots+(n+1) = \frac{(n+1)((n+1)+1)}{2}\).
Show that \(1+2+\dots+n = \frac{n(n+1)}{2}\) for every natural number \(n\).
Show that if \(1+2^1+2^2+\dots+2^{10} = 2^{11} - 1\), then \(1+2^1+2^2+\dots+2^{11} = 2^{12} - 1\).
Show that \(1+2^1+2^2+\dots+2^n = 2^{n+1} - 1\) for every natural number \(n\).
What is wrong with the following proof that “all rulers have the same length" using induction?
Base case: suppose that we have one ruler, then clearly it clearly has the same length as itself.
Assume that any \(n\) rulers have the same length for the induction hypothesis. If we have \(n+1\) rulers, the first \(n\) ruler have the same length by the induction hypothesis, and the last \(n\) rulers have the same length also by induction hypothesis. The last ruler has the same length as the middle \(n-1\) rulers, so it also has the same length as the first ruler. This means all \(n+1\) rulers have the same length.
By the principle of mathematical induction, all rulers have the same length.
The AM-GM inequality asserts that the arithmetic mean of nonnegative numbers is always at least their geometric mean. That is, if \(a_1,\dots,a_n\geq 0\), then \[\frac{a_1+\dots+a_n}{n}\geq \sqrt[n]{a_1\dots a_n}.\] Prove this inequality.
There are many proofs of this fact and quite a few of them are by induction. In fact, one of the most creative uses of induction can be found in Cauchy’s proof of the AM-GM inequality in Cours d’analyse.
Consider the \(4!\) possible permutations of the numbers \(1,2,3,4\). Which of those permutations keep the expression \(x_1x_2+x_3x_4\) the same?
Prove that for all positive integers \(n\) there exists a partition of the set of positive integers \(k\le2^{n+1}\) into sets \(A\) and \(B\) such that \[\sum_{x\in A}x^i=\sum_{x\in B}x^i\] for all integers \(0\le i\le n\).
Find all solutions to \(x^2+2=y^3\) in the natural numbers.