Sequences+and+Series

A **sequence** is another word for a list, for example, all of the following are sequences of numbers. 1, 2, 3, 4, 5, 6, … 5, 8, 11, 14, 17, … 2, 4, 8, 16, 32, … 1, 1, 2, 3, 5, 8, … 3, 1, 4, 1, 5, 9, 2, …

These sequences should all look familiar to you. We refer to the first two as **arithmetic sequences**. The important characteristic that makes these sequences arithmetic is that the difference between each term is the same, the **common difference**. Problem 1: If k+1, 2k, and k + 7 are three consecutive terms of an arithmetic sequence, what is k?

Solution: Since the sequence is arithmetic, we know that the difference between the first and the second term must be the same as the difference between the second and third terms.

2k – (k + 1) = (k+7) – 2k

k – 1 = 7 – k

2k = 8

k = 4

giving the sequence 5, 8, 11 which is certainly arithmetic, with a common difference of 3.

In general we write the nth term of an arithmetic sequence

where //**d**// is the common difference and //**u1**// the first term of the sequence. Once you know these two values you should be able to find any term of an arithmetic sequence. But what if we want to sum the terms of a sequence?

For example what is 1 + 3 + 5 + 7 + 9 + 11 + … + 101?

The sum of the terms gives us an **arithmetic series**, and fortunately we do not have to add these terms one-by-one. The story goes that it was a problem such as this, in fact summing the first 100 integers that was set to Carl Friedrich Gauss, aged 8, in order for his teacher to get a bit of a break from teaching. The young Gauss responded with the answer almost immediately.

Consider the general arithmetic series 

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">Or, since we know each term is simply the last one plus the common difference

<span style="display: block; font-family: Arial,Helvetica,sans-serif; line-height: 0px; overflow-x: hidden; overflow-y: hidden; text-align: justify;">

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">Let us call this sum Sn

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">We might observe that it does not matter in which order we sum these terms, so similarly

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;"> <span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">Now let us add the previous two lines.

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">We note that each ‘pair’ of terms give the same sum.

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">and so <span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">instead of the common difference, we could have thought of the sums of terms as the first term + the last term. Using l for the last term would give the following formula. <span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">This may be more appropriate to use, depending on the situation.

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">A **geometric sequence** is one which we find the next term by multiplying the previous one by a particular value, we say the sequence has a **common ratio**, for example in the sequence below 3 is the common ratio.

<span style="font-family: Arial,Helvetica,sans-serif;">1, 3, 9, 27, 81, …

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">As with arithmetic sequences we want an expression for a general term of the sequence, which is

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">where **//r//** is the common ratio.

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">And, as with arithmetic sequences if we sum the terms of a geometric sequence we have a //**geometric series**//. Below is the proof of the general formula for the sum of the first n terms of a geometric series. Read through it and ensure you understand each step.













<span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">or similarly

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">Note that in the last step we divide by //**(r-1)**//. This is not a problem unless r = 1, so we should add



<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">Problem 2. What is the sum of the first ten terms of the following series?

<span style="font-family: Arial,Helvetica,sans-serif;">1 + 2 + 4 + 8 + 16 + …

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">Solution 2. The first term in the series is 1 and the common ratio is 2, so applying the above formula





<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">Problem3. What is the sum of the first ten terms of the following series?

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">

<span style="display: block; font-family: Arial,Helvetica,sans-serif; text-align: justify;">Solution 3. The first term in the series is 1 and the common ratio is 1/2, so applying the above formula





It might appear to us in this second case that the sum is very close to 2, and we may ask ourselves what if we summed the first 100 terms or the first thousand terms or perhaps what if we summed all the terms in this infinite sequence?

Let us consider what happens if we imagine doing this, we will use the symbol and refer to it as the ‘sum to infinity’ of the sequence. Using the formula.



But is a bit of an odd thing to write, and we can’t really think of infinity as a number. Instead what we are really considering is what happens to this value as we tend towards infinity, and the answer is that it tends to zero, giving the following.





We note that whenever the common ratio is sufficiently small, in fact whenever **|r| < 1** the **rn** bit in the formula will tend to zero as n tends to infinity. Giving the following formula.