Sequences & Series

Arithmetic Series

In arithmetic sequences we considered the example of an amphitheater with 3×5=15 seats in the first row, 3×7=21 in the second, 3×9=27 and so on.

We therefore concluded that there were

u9=u1+8d=15+8⋅6=63

seats in the 9th row.

But here's a more interesting question, how many seats are there in total, combining all the rows 1 through 9?

Discussion

Visualizing and counting the number of seats in this large amphitheater is tricky. Can you come up with an example of a simpler arithmetic sequence that be easier to think about?

To simplify things a little bit, let's consider the sequence

3,4,5,6…

which is arithmetic with u1=3 and d=1.

Checkpoint

What is the sum of the first 4 terms in the sequence 3,4,5,6…?

Select the correct option

A series is the sum of a sequence

The sum of terms in a sequence is called a series.

In the case of the sequence

3,4,5,6…

we can construct the series

3+4+5+6+⋯

Because the series continues forever, we write Sn to denote the sum of the first n terms in the series.

For the series 3,4,5,6… this is

Sn=3+4+⋯+(n−1)+n

Now imagine I want to find the sum of the first 10 terms in this sequence, that is

3+4+5+⋯+10

I can visualize this using dots:

Using the diagram above, can you calculate the total number of dots, without calculating manually?

Discussion

To make this more visually intuitive, imagine taking these dots and reflecting them to make a rectangle:

How many dots are there in the rectangle? How many are red?

The rectangle is 8 dots wide and 13 dots high, so it contains

8×13=104

dots in all.

Reflecting it to create the rectangle doubled the total number of dots, so half of all the dots must be red, that is

21(104)=52

Thus there are 104 dots in total, of which 52 are red.

Let's break that down to illustrate how we can turn this into a formula for any arithmetic series:

# of red dots =28×(3+10)

The essence of this formula is recognizing that the terms form pairs:

u1+u8u2+u7u3+u6u4+u5=3+10=13=4+9=13=5+8=13=6+7=13

Each pair adds up to 13, the sum of the first and last pair! And how many pairs are there? 28.

So that's how we got the formula

# of red dots =28×(3+10)

Let's now imagine any arithmetic sequence with first term u1 and common difference d. The sum of the first n terms can be found by pairing terms:

u1+un=u2+un−1

This works because u2=u1+d, and un−1=un−d, so

u2+un−1=u1+d+un−d=u1+un

And there will be 2n pairs, so the sum of first n terms is

Sn=2n(u1+un)

where Sn means "the sum of the first n terms".

Exercise

An arithmetic sequence has first term 7 and 20th term 67. Find the sum of the first 20 terms.

Select the correct option

Discussion

Recall the formula for un that we found in the previous lesson. Can you combine that formula with the new formula Sn=2n(u1+un) to find another formula for Sn?

The formula for the nth term of an arithmetic sequence is

un=u1+(n−1)d

The sum of the first n terms is

Sn=2n(u1+un)

Substitute the formula for un into the sum formula:

Sn=2n(u1+[u1+(n−1)d])

Combine like terms inside the brackets:

Sn=2n(2u1+(n−1)d)

This gives an alternative formula for the sum of the first n terms of an arithmetic sequence.

Calculating arithmetic series

The sum of the first n terms in an arithmetic sequence is given by

Sn=2n(2u1+(n−1)d)📖

or equivalently

Sn=2n(u1+un)📖

Exercise

An arithmetic sequence has first term 7 and common difference -5. Find the sum of the first 10 terms.

Select the correct option

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