Lesson Summary - 1.5 PDF

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Colin O’Brien 9/4/17 Lesson Summary: 1.5 In Lesson 1.5 - Don’t Break the Chain we solved problems around the situation about a chain email. In it, Bill Weights, the founder of Super Scooper Ice Cream sends out a chain email offering a discount if you forward the chain email to ten people. Bill starts the chain by sending the email to 8 of his close friends. Assuming that everyone who receives it forwards the email and that it takes those recipients a day to forward the email, we did different things such as create an equation for the situation or solve how much money Bill Weights spent. The explicit equation that my team created was f(n) = 8 x 10(n-1)  and the recursive equation was f(n) = f(n-1) x 10. In this situation, n = Time (days) since Bill first sent out the email. The first important concept that we covered in the lesson is comparing equivalent equations and which are better for a certain situation. For example, in the chain email situation, two equations that we compared in class were f(n) = 8 x 10(n-1) and f(n) = 0.8(10)n . Both equations are equivalent and work for the situation however the first one, f(n) = 8 x 10(n-1)  is a better representation of this situation. This equation better represents the situation because different parts of the equation

show different parts of the situation. The “8” represents the first 8 friends that Bill Weights sent the email to to start the chain. “x 10(n-1)  ” represents how much the total number of forwards grows by. Each day the total number of emails sent out is ten times more than the day before. This is why there’s the exponent, “(n-1)  ” because you take the total number of days, “n” and then subtract 1, “-1” to get the day before which it is then ten times more of. The other important part of this lesson is comparing geometric and arithmetic sequences in the form of recursive and explicit equations. When talking about recursive equations, if the sequence is either geometric or arithmetic you there will most likely be the term, “f(n - 1)”. The other similarity between the two types of sequences is that unless specified, both will most likely give you a starting term of, “f(1) = x” where x could equal anything. The main difference between the two is that after, “f(n - 1)” in a geometric sequence you will multiply by something but in an arithmetic sequence you will add something to the previous term. For example, in the the geometric sequence, {2,4,8,16…} the recursive equation would be f(n) = f(n-1) x 2 and f(1) = 2. In the arithmetic sequence, {2,4,6,8…} the recursive equation would be f(n) = f(n - 1) + 2 and f(1) = 2. Notice that both equations have “f(1) = something” as their starting term. Both equations also have the term, “f(n - 1)”. Also notice that the geometric equation has a multiplying 2

while the arithmetic equation has an adding 2. When you compare explicit equations of geometric and arithmetic sequences you’ll notice that both equations represent the starting value in the equations themselves. The main difference you’ll see is that geometric equations always have variables as exponents and arithmetic equations always have a multiplication of a variable. For example, if we made explicit equations for the two sequences we used earlier {2,4,8,16…} and {2,4,6,8…} the geometric equation would be f(n) = 2 x 2(n - 1) and the arithmetic equation would be f(n) = 2(n - 1) + 2. Notice that in both equations they show the starting value of 2 in both equations. In the geometric it’s the, “2 x” and in the arithmetic it’s the, “+ 2”. Also, notice that the geometric equation has the exponent of, “(n-1)  ” whereas the arithmetic equation has the multiplier of, “(n - 1)”.While they do different things both of the, “n”s represent the stage you are currently on. Lastly, one important vocabulary word we learned during this lesson is binomial. When something is binomial it is describing something that has two terms. For example a binomial expression might be 2n + 1. The two terms would be, “2n” and, “1”....