High School: Algebra
Seeing Structure in Expressions HSA-SSE.B.4
4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
Although students may not know it, they already know a lot about sequences and series. Not only are sequences and series on TV, in music, and all over the Internet, they were also taught between nap time and making those dried macaroni picture frames. You could have a sequence as easy as one, two, three.
In mathematics, a sequence is a bunch of numbers listed one after the other. In many cases, it is possible to determine a particular member of a sequence simply from its location, just like when counting natural numbers. That sequence starts with the number 1 and every other member of the sequence can be found by taking the previous member of the sequence and adding 1 to it.
Students should know the difference between an arithmetic sequence and a geometric sequence. In arithmetic sequences, successive members have a common difference. That is, the difference between each member and the one before it is some constant value. In geometric sequences, all numbers have a common ratio, meaning that the quotient of each member and the one before it is some constant value.
It's best to give several examples of both arithmetic and geometric sequences and series, and then finally give a formula for each. For instance, if a is the first term in an arithmetic series and the common difference is d, then the series would be the sum of a, a + d, a + 2d, a + 3d, and so on. For a geometric sequence with common ratio r, we'll have a, ar, ar2, ar3, ar4, ar5, and so on.
Notice that the first member of the sequence has no factors of r. The second member has 1 factor of r, the third has 2 factors of r, the fourth has 3 factors of r, and so on. The number of factors of r is always one less than the number of that particular term in the sequence. If n represents the number of the nth member of the sequence, then that number has a value of arn – 1.
Students should know that a series is the sum of the members of a sequence. We can form sums of both arithmetic and geometric sequences. For an n-term geometric sequence, we can write the series as sn = a + ar + ar2 + ar3 + ar4 + ar5 + ... + arn – 1.
This is all fine if we have 6 or 7 terms, but imagine trying to use this to calculate a series with 100 terms. Or 200 terms. Or an infinite number of terms. That would be an awful, not to mention time-consuming calculation to do by hand. Luckily, we don't have to, because we can perform a little magic to greatly simplify this expression. (It's actually not magic, but don't tell your students that. Some of them are still waiting for their invitation to Hogwarts, and this is the closest they'll ever get.)
Let's take the expression for the series and multiply it by the common ratio, r.
rsn = ar + ar2 + ar3 + ar4 + ar5 + ar6 + ... + arn
Notice that the number of factors of r in each term in the sum increased by 1. Now, let's take our original expression for s and subtract this new expression from it. There is nothing that says we can't do that, right?
sn – rsn = (a + ar + ar2 + ar3 + ar4 + ar5 + ... + arn – 1) – (ar + ar2 + ar3 + ar4 + ar5 + ar6 + ... + arn)
Notice that except for the very first term in the expression for s and the very last term in the expression for rs, there are matching terms in the two sets of parentheses. That means all of those terms cancel each other out, and we're left with sn – rsn = a – arn.
We can then factor the left hand side of that equation as sn(1 – r). Dividing both sides by 1 – r and factoring out the a in the numerator on the right side gives us a much simpler way to find the sum of a geometric series.
Your students should be applauding right about now. We know it's tough to compete with Hogwarts, but come on. This math is clearly not meant for mere muggles.
We do have to be careful about one thing, though. Notice that we divide by 1 – r. If r = 1, we have a serious problem since that's dividing by 0. So we have to place a restriction on this formula, since using it when r = 1 might bring back He-Who-Must-Not-Be-Named...again.
Geometric series appear in many places, but one case which touches just about everyone is in the area of finance and economics. Calculations involving loan payments (for a mortgage, for instance) often make use of geometric series.
Suppose you took out a $200,000 loan for a dragon. It sounds expensive, but it's worth it if you're a frequent flyer. Plus, it's a dragon.
The beastly $200,000 loan has a 30-year fixed interest rate of 3.6% per year paid monthly. By convention, this means that each month, an interest rate of 0.036 ÷ 12 = 0.003, or 0.3% will be applied to the unpaid principal to calculate the interest payment for that month. This will continue for each month until the term of 30 years has been reached, which is (12)(30) = 360 months. If m is the monthly amount, P is the principal amount on the loan, r is the rate, and n is the number of months, we can use the following formula to find the monthly amount.
There are calculators online which will do this sort of calculation for you, but it is nice to understand how the calculation is done so your students can verify it for themselves. As far as dragons go, nobody wants to pay more than they have to. On second thought, that probably applies to just about anything.