Hello! I have a lame trick for you. Think of a number, which we call the original number from now on. Do the following:
Add fifteen to the original number.
Multiply the resulting number by four.
Add eight times the original number to the new result.
Divide by six.
Subtract twice the original number.
But I already know the finally answer! How?
Calculate the left side and the right side. \[2\times(12+3)\quad\quad 2\times 12 + 2\times 3\] \[3\times(0.8+1)\quad\quad 3\times 0.8 + 3\times 1\] \[(-2)\times (3-5)\quad\quad (-2)\times 3 + (-2)\times (-5)\] What do you notice?
Show that for any number \(a,b,c,d\), we have \((a+b)(c+d) = ac + ad + bc + bd\).
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.