![]() If r = 1, we have a serious problem since that's dividing by 0. We do have to be careful about one thing, though. This math is clearly not meant for mere muggles. We know it's tough to compete with Hogwarts, but come on. Your students should be applauding right about now. 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. ![]() We can then factor the left hand side of that equation as s n(1 – r). That means all of those terms cancel each other out, and we're left with s n – rs n = a – ar n. 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. S n – rs n = ( a + ar + ar 2 + ar 3 + ar 4 + ar 5 +. There is nothing that says we can't do that, right? Now, let's take our original expression for s and subtract this new expression from it. Notice that the number of factors of r in each term in the sum increased by 1. Rs n = ar + ar 2 + ar 3 + ar 4 + ar 5 + ar 6 +. Let's take the expression for the series and multiply it by the common ratio, r. Some of them are still waiting for their invitation to Hogwarts, and this is the closest they'll ever get.) (It's actually not magic, but don't tell your students that. Luckily, we don't have to, because we can perform a little magic to greatly simplify this expression. That would be an awful, not to mention time-consuming calculation to do by hand. This is all fine if we have 6 or 7 terms, but imagine trying to use this to calculate a series with 100 terms. ![]() For an n-term geometric sequence, we can write the series as s n = a + ar + ar 2 + ar 3 + ar 4 + ar 5 +. We can form sums of both arithmetic and geometric sequences. Students should know that a series is the sum of the members of a sequence. If n represents the number of the nth member of the sequence, then that number has a value of ar n – 1. The number of factors of r is always one less than the number of that particular term in the sequence. 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. Notice that the first member of the sequence has no factors of r. For a geometric sequence with common ratio r, we'll have a, ar, ar 2, ar 3, ar 4, ar 5, and so on. 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 + 2 d, a + 3 d, and so on. It's best to give several examples of both arithmetic and geometric sequences and series, and then finally give a formula for each. 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. That is, the difference between each member and the one before it is some constant value. In arithmetic sequences, successive members have a common difference. Students should know the difference between an arithmetic sequence and a geometric sequence. 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. In many cases, it is possible to determine a particular member of a sequence simply from its location, just like when counting natural numbers. In mathematics, a sequence is a bunch of numbers listed one after the other. You could have a sequence as easy as one, two, three. 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. For example, calculate mortgage payments.Īlthough students may not know it, they already know a lot about sequences and series. 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. That is, each subsequent term is found by multiplying the previous term by the common ratio.Īs for the sum of these progressions it is best to remember how to find the sums rather than to memorize formulas.4. Whereas the arithmetic sequence has a common difference, d, between the terms, a geometric sequence has a common ratio, r. Ī geometric sequence, also called a geometric progression, also begins with a fixed number, a, and then each subsequent term is found by multiplying by a constant value, r, called the common ratio. General Form: a, a + d, a + 2d, a + 3d +. What are the equations for geometric and arithmetic sequences?Īlso, what are the equations for finding the sums of those series?Īn arithmetic sequence, also called an arithmetic progression, is a sequence that begins with a fixed number, a, and then each subsequent term is found by adding a constant value, d, called the common difference.
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