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Sequences and series are introduced in many early Algebra courses, but come back in a major way during a Calculus course.

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A sequence is simply a list of things, which are usually numbers, in some sort of order.

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For example, we could have the sequence 2, 4, 6, 8, 10, …, showing us that our sequence is increasing even numbers. We could also view this as simply adding 2 to each previous term in the sequence.

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There are two kinds of sequences to be aware of:

- Arithmetic
- Geometric

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Arithmetic sequences involve adding or subtracting the same value to go from one term to the next.

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This is like the sequence example that we saw earlier, but we could also consider the decreasing sequence 20, 15, 10, 5, 0, -5, ….

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This sequence has a common difference of -5, since that is what we add to each term. Note that the increasing even numbers sequence from earlier has a common difference of 2.

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To find the nth term of a sequence, we need to take the first term and then add the common difference a certain number of times.

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The formula *a*_{n} = *a*_{1} + (*n* - 1)*d* where a_{n} is the nth term, *a*_{1} is the first term, *n* is which term we are looking for, and *d* is the common difference is particularly useful.

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Using the sequence 2, 4, 6, 8, … we can calculate the 5th term using the formula!

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This shows us the following set of equations:

- a
_{5 }= 2 + (5-1)2 - a
_{5 }= 2 + (4)2 - a
_{5 }= 2 + 8 = 10

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We also could have just added another 2 to the sequence, but this formula works for very large values of n. We could have even found the 100th term using this method!

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Geometric sequences involve multiplying or dividing the same value to go from one term to the next.

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Just like arithmetic sequences, we can list off a sequence with a common ratio of 2 as follows:

1, 2, 4, 8, 16, …

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In this case, we simply double each term to get to the next.

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We can also halve each term if the common ratio is ½.

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This would look like the following sequence:

128, 64, 32, 16, …

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To calculate the nth term of a geometric sequence, you can use the following formula: *a _{n}* =

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To find the 5th term of the 128, 64, 32, 16… sequence, we can plug in our values to the formula.

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This gives

- a
_{5}= 128(1/2)^{5-1} - a
_{5}= 128(1/2)^{4 }= 128(1/16) = 8

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Again, we could have just divided 16 by 2 to find the next term is 8, but where’s the fun in that?

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A series is simply the sum of a sequence.

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Series can also be arithmetic or geometric.

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An arithmetic series could look like 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8. Notice that it doesn’t need to go off to infinity, the same is true for sequences, which can end whenever you want!

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We could simply add up the values from left to right, but there is a more clever solution. Pair up the first and last number 1 + 8 = 9. Take the second from the start and the second from the last number to see that 2 + 7 = 9. How many pairs like this do we have?

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There are 8 numbers, so we have 8/2 = 4 pairs!

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Each pair adds up to 9, so the total sum is 9 × 4 = 36.

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This idea can be generalized in the following formula:

*S _{n}* =

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Notice that *n*/2 exists because that’s the number of pairs and the first term plus the last term is just what each pair adds up to.

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Geometric series are just the sum of a geometric sequence and are almost always found for a decreasing geometric sequence that has an infinite number of terms.

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A series that fits this description is:

4 + 2 + 1 + ½ + ¼ + …

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The formula for this sum is: S_{∞} = *a*_{1} / 1-*r*

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Therefore the infinite geometric series above has a sum of , since the common ratio is 1/2 and our first term is 4.

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In summary, sequences are simply an ordered set of things, while a series is a sum of those things.

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The formulas are incredibly useful, but the ideas behind them are even more important and you’ll never have to simply memorize them if you understand where they come from!

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