M3U1D3 Warm Up 12, 6, 0, -6, -12 12, 4, 4/3, 4/9, 4/27 2, 5, 8, 11, 14 0, 2, 6, 12, 20 Arithmetic an = an-1 + 4 Geometric an = 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.” 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 . . . 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. 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.

Easy Cheesy! But isn’t there a quicker way to do this??? Evaluating a Series To 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 Where: Sn is the sum of all the terms n = number of terms a1 = first term an = last term Just like yesterday!!! From our last example: 2+11+20+29+38+47 = 147 147 Let’s try one: evaluate the series: 5, 9, 13,17,21,25,29

WRITE THESE FORMULAS DOWN ON YOUR HINTS CARD!!! Vocabulary of Arithmetic Sequences and Series (Universal) 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 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.

UPPER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE) LOWER BOUND (NUMBER)

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) Explicit function for the sequence Summation symbol 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 3: 102 + 104 + 106 + 108 + 110 + 112 n = 6 terms 1st term = 102 Rule: Hmmmm. . . . Rule = 100 + 2n Yes, you can add manually. But let’s try using the shortcut: Let’s Evaluate:

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? Note: this is NOT an arithmetic series. You can NOT use the shortcut; you have to manually crunch out all the values. N = 10 1st = 1 Last = 10 a1= 1-3 = -2 a10 = 10-3 = 7 Why 4? N = 4 1st = 2 Last = 5 4+9+16+25 = 54 Notice we can use the shortcut here:

To find an, substitute & Evaluate!!! Ex 4: -19 353 ?? 63 x 6 To find an, substitute & Evaluate!!!

Compound Interest REVISED Compound interest formula: A = the compound amount or future value P = principal i = interest rate per period of compounding n=number of times compounded per year t = number of years I = interest earned

Distribute Project directions and rubric DUE Sept. 15th !!!

U1D3 Arithmetic Series and Summation Notation WS #1-16 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.