Question 3. Summation Notation and Product Notation. Sometimes, when performing calculations, we have to add an entire list of numbers or multiply an entire list of numbers. It's helpful to have a concise notation for doing so. More concretely, let $1,:c2,:c3, - - be a list of numbers. If we want to add together the rst T numbers on this list, we can use the following summation notation: T Em, Em +:c2+---+a:T_1+:cT. i=1 If we want to sum the nth through Tth numbers on the list, we could write: T 2332: E$n+$n+1+\"'+3T71 +5131"- t=n Similarly, if we want to multiply the rst T numbers on the list, we can use the following product notation: T mtE$1XJ2XHX$Tn1XIETu 1 Mathematically, there's nothing new here; we're just performing addition and multiplication. However, the notation may be new to some students, so the following questions are meant to acclimate you to using it. Below, I'll ask you to show a couple of expressions to be true. Again, this entails providing the algebraic steps to justify why the left-hand side of an equation is equal to the right-hand side. a) Let m0,m1,m2,- -- be a list of numbers. Show that T 113T = 130 + Z Amt: t=1 where A is the change in a: . For example, if :0, stands for the amount of money circulating in the economy, we can think of mg as the initial stock of money, and Ax, is the amount of new money created at date t. In that case, the above equation says that the amount of money circulating at date T is equal to the initial stock of money, plus the amount of money created in each period up through date T . b) Let 130,131,332, - -- be a list of numbers. Show that: T pr = 100 X H(1 + 70AM): t=1 where %A is the percentage change. Recall that, if pt stands for the price of goods, then %Apt is the ination rate. Consequently, the above equation says that the price of goods at date T equals the initial price level, times the ination rate compounded over each period up through date T