Transcript 11.2 – Arithmetic Sequences

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 . . .
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.
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.
Note: if the numbers go on forever, it is infinite; if it has a
definitive ending it is finite.
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
n
Sn  ( a1  an )
2
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
Sn 
6
( 2  47)  147
2
Let’s try one: evaluate the series:
5, 9, 13,17,21,25,29
Sn 
7
(5  29)  119
2
Summation
When 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.
Upper Limit (greatest value of n)
Summation symbol
Explicit function for
the sequence
Lower Limit (least value of n)
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:
Sn 
n
( a1  an )
2
Yes, you can add manually. But let’s try
using the shortcut:
Sn 
6
(102  112)  642
2
Let’s try some
Find 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:
Why is the
answer not 58?
N = 10
1st = 1
Notice we can use the
shortcut here:
Last = 10
a1= 1-3 = -2
N=4
1st = 2
Last = 5
a10 = 10-3 = 7
4+9+16+25 = 54
Sn 
10
( 2  7)  5(5)  25
2
Note: this is
NOT an
arithmetic
series. You can
NOT use the
shortcut; you
have to
manually
crunch out all
the values.