Sequences and series are introduced in many early Algebra courses, but come back in a major way during a Calculus course.
A sequence is simply a list of things, which are usually numbers, in some sort of order.
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.
There are two kinds of sequences to be aware of:
Arithmetic sequences involve adding or subtracting the same value to go from one term to the next.
This is like the sequence example that we saw earlier, but we could also consider the decreasing sequence 20, 15, 10, 5, 0, -5, ….
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.
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.
The formula an = a1 + (n - 1)d where an is the nth term, a1 is the first term, n is which term we are looking for, and d is the common difference is particularly useful.
Using the sequence 2, 4, 6, 8, … we can calculate the 5th term using the formula!
This shows us the following set of equations:
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!
Geometric sequences involve multiplying or dividing the same value to go from one term to the next.
Just like arithmetic sequences, we can list off a sequence with a common ratio of 2 as follows:
1, 2, 4, 8, 16, …
In this case, we simply double each term to get to the next.
We can also halve each term if the common ratio is ½.
This would look like the following sequence:
128, 64, 32, 16, …
To calculate the nth term of a geometric sequence, you can use the following formula: an = a1(r)n-1
To find the 5th term of the 128, 64, 32, 16… sequence, we can plug in our values to the formula.
Again, we could have just divided 16 by 2 to find the next term is 8, but where’s the fun in that?
A series is simply the sum of a sequence.
Series can also be arithmetic or geometric.
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!
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?
There are 8 numbers, so we have 8/2 = 4 pairs!
Each pair adds up to 9, so the total sum is 9 × 4 = 36.
This idea can be generalized in the following formula:
Sn = n/2 (first term + last term)
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.
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.
A series that fits this description is:
4 + 2 + 1 + ½ + ¼ + …
The formula for this sum is: S∞ = a1 / 1-r
Therefore the infinite geometric series above has a sum of , since the common ratio is 1/2 and our first term is 4.
In summary, sequences are simply an ordered set of things, while a series is a sum of those things.
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!