Expand \((x_1+\dots + x_n)^2\) where \(x_1,\dots,x_n\) are real numbers.
Prove the Cauchy-Schwarz inequality \[(a_1b_1+\dots+a_nb_n)^2\leq (a_1^2+\dots+a_n^2)(b_1^2+\dots+b_n^2)\] where \(a_1,\dots,a_n,b_1,\dots,b_n\) are real numbers. If you already know a proof (or more!), find a new one.
Show that the sum of any \(100\) consecutive numbers is a multiple of \(50\) but not a multiple of \(100\).
Alice sums \(n\) consecutive numbers, not necessarily starting from \(1\), where \(n\) is a multiple of four. An example of such a sum is \(5+6+7+8\). Can this sum ever be odd?
Show that the difference between two consecutive square numbers is always odd.
One of the most powerful ideas in mathematics is that we can use letters — like \(a, b, n\) or \(x\) — to stand for numbers, shapes or other things. When we do this, we can reason about all such possible objects at once, without knowing exactly which number or shape we are really dealing with.
For example, the statement \[\text{``Let } a,b \text{ be numbers. Then } a+b=b+a."\] is true no matter what numbers \(a\) and \(b\) are. It tells us all of the following at the same time: \[3+5=5+3, \qquad (-10)+(-2)=(-2)+(-10), \qquad 7+0=0+7,\] and many more.
The rule \(a+b=b+a\) does not depend on the “three-ness” of \(3\) or the “five-ness” of \(5\) — it works for any numbers. Of course, we could not let \(a\) be a triangle and \(b\) be a tiger, because we do not know what it means to add a triangle to a tiger! Our rules only apply to objects for which the operations make sense.
This way of using symbols to express rules and patterns is what we call algebra. As long as we follow the rules that numbers follow, our reasoning will stay true. Today we will practise using these symbols to work with the algebra of numbers — it may take effort, but it is an important skill that will help you a lot in your mathematical journey.
Let \(n\) be a natural number and \(x=2n^2+n\). Prove that the sum of the square of the \(n+1\) consecutive integers starting at \(x\) is the sum of the square of the \(n\) consecutive integers starting at \(x+n+1\).
For example, when \(n=2\), we have \(10^2+11^2+12^2=13^2+14^2\)!