11.4 – Arithmetic Series

How do I know if it is an arithmetic series?

• A series is the expression for the sum of the terms of a sequence, not just “what is the next terms

Ex: 6, 9, 12, 15, 18 . . .

This is a list of the numbers in the pattern an not a sum. It is a sequence. Note it goes on forever, so we say it is an infinite sequence.

Ex: 6 + 9 + 12 + 15 + 18

Note: if the numbers go on forever, it is infinite; if it has a definitive ending it is finite.

Here we are adding the values. We call this a series. Because it does not go on forever, we say it is a finite series.

Evaluating a Series

• Simply add up the values.

Ex: 2, 11, 20, 29, 38, 47

2+11+20+29+38+47 = 147

Easy Cheesy! But isn’t there a quicker way to do this???

Sum of a Finite Arithmetic Series

)(2

1 nn aan

S

)472(2

6nS

Where: Sn is the sum of all the terms

n = number of terms

a1 = first term

an = last term

From our last example: 2+11+20+29+38+47 = 147

119)295(2

7nS

147

Let’s try one: evaluate the series:

5, 9, 13,17,21,25,29

SummationWhen we don’t want to write out a whole bunch of numbers in the series, the summation symbol is used when writing a series. The limits are the greatest and least values of n.

Summation symbol

Upper Limit (greatest value of n)

Lower Limit (least value of n)

Explicit function for the sequence

So, the way this works is plug in n=1 to the equation and continue through n=3.

(5*1 + 1) + (5*2 +1) + (5*3 + 1) = 33

Writing a series in summation form

• Ex: 102 + 104 + 106 + 108 + 110 + 112

n = 6 terms

1st term = 1

Rule: Hmmmm. . . .

Rule = 100 + 2n

Let’s Evaluate: Yes, you can add manually. But let’s try using the shortcut:

)(2

1 nn aan

S 642)112102(2

6nS

Let’s try someFind the number of terms, the first term and the last term.

Then evaluate the series:

Let’s try some

Find the number of terms, the first term and the last term. Then evaluate the series:

N = 10

1st = 1

Last = 10

a1= 1-3 = -2

a10 = 10-3 = 7

N = 4

1st = 2

Last = 5

4+9+16+25 = 54

Notice we can use the shortcut here:

Why is the answer not 58?

Note: this is NOT an arithmetic series. You can NOT use the shortcut; you have to manually crunch out all the values.25)5(5)72(

2

10nS