M3U1D3 Warm Up

12, 6, 0, -6, -1212, 4, 4/3, 4/9, 4/27

2, 5, 8, 11, 140, 2, 6, 12, 20

Arithmetican = an-1 + 4

Geometrican = a1(1/2)n-1

Homework Check:

Homework Check:

M3U1D3 Arithmetic Series

Objective:To write arithmetic and geometric

sequences both recursively and with an explicit formula, use them to model situations, and translate between the

two forms. F-BF.2

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 are 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 SeriesTo evaluate a series, simply add up the values!

Ex 1: 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)(

21 nn aa

nS

)472(2

6nS

Where: Sn is the sum of all the terms

n = number of terms

a1 = first term

an = last termFrom 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

Just like yesterday!!!

Vocabulary of Arithmetic Sequences and Series (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

n 1

n 1 n

nth term of arithmetic sequence

sum of n terms of arithmetic sequen

a a n 1 d

nS a a

2ce

WRITE THESE FORMULAS DOWN ON YOUR HINTS CARD!!!

Example 2:Find the sum of the first 50 terms of an arithmetic series with a1 = 28 and d = -4

We need to know n, a1, and a50.

n= 50, a1 = 28, a50 = ?? We have to find it.

Example 2 CONT.

a50 = 28 + -4(50 - 1) = 28 + -4(49) = 28 + -196 = -168

So n = 50, a1 = 28, & a50 =-168

S50 = (50/2)(28 + -168) = 25(-140) = -3500

To write out a series and compute a sum can sometimes be very tedious. Mathematicians often use the Greek letter sigma & summation notation to simplify this task.

B

nn A

a

UPPER BOUND(NUMBER)

LOWER BOUND(NUMBER)

SIGMA(SUM OF TERMS) NTH TERM

(SEQUENCE)

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 formEx 3: 102 + 104 + 106 + 108 + 110 + 112

n = 6 terms

1st term = 102

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:

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

Why 4?

63Find S of 19, 13, 7,...

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-19

63

??

x

6

n 1a a n 1 d

?? 19 6 1

?? 353

3 6

353

n 1 n

nS a a

2

63

633 3S

219 5

63 1 1S 052

To find an, substitute & Evaluate!!!

Ex 4:

Compound Interest REVISEDCompound interest formula:

A = the compound amount or future valueP = principali = interest rate per period of compounding

n=number of times compounded per yeart = number of yearsI = interest earned

PAIandniPA nt )/1(

ProjectDistribute Project

directions and rubric

DUE Sept. 15th !!!

Classwork:U1D3 Arithmetic Series and

Summation Notation WS #1-16

Homework:U1D3 WS2 #1-14

AND have your parent read the class

letter then sign & return the actual Math III acknowledgement

sheet to me.