M3U1D2 Warmup: Find the next three terms of each sequence. Explain how you found the terms. 1.2, 5, 8, 11, ______, ______, ______ 2. 12, 7, 2, -3, ______, ______, ______ Added 3 each time Added -5 each time.

Homework Check: Document Camera Collect Parent Letters

M3U1D2 Sequences: Arithmetic and Geometric 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

A sequence in which a constant (d) can be added to each term to get the next term is called an Arithmetic Sequence. The constant (d) is called the Common Difference. To find the common difference (d), subtract any term from one that follows it a1a1 a2a2 a3a3 a4a4 a5a5

Find the first term and the common difference of each arithmetic sequence. First term (a):4 Common difference (d):= 9 – 4 = 5 First term (a): 34 Common difference (d): -7 BE CAREFUL: ALWAYS CHECK TO MAKE SURE THE DIFFERENCE IS THE SAME BETWEEN EACH TERM !

Now you try! Find the first term and the common difference of each of these arithmetic sequences. b) 11, 23, 35, 47, …. a) 1, -4, -9, -14, …. a = 1 and d = a 2 - a 1 = = - 5 a = 11 and d = a 2 - a1 a1 = = 12

The first term of an arithmetic sequence is (a 1 ). We add (d) to get the next term. There is a pattern, therefore there is a formula we can use to find any term that we need without listing the whole sequence. The nth term of an arithmetic sequence is given by: The last # in the sequence/or the # you are looking for First term The position the term is in The common difference a n = a 1 + (n – 1) d Is this Explicit or Recursive?Explicit!!!

Find the 14 th term of the arithmetic sequence 4, 7, 10, 13,…… a n = a 1 + (n – 1) d a 14 = You are looking for the term! The 14 th term in this sequence is the number 43!

Now you try! Find the 10th and 25 th term given the following information. Make sure to derive the general formula first and then list what you have been provided. b) x+10, x+7, x+4, x+1, …. a) 1, 7, 13, 19 …. d) The second term is 8 and the common difference is 3 c) The first term is 3 and the common difference is -21

b) x+10, x+7, x+4, x+1,. a) 1, 7, 13, 19 …. …. d) The second term is 8 and the common difference is 3 c) The first term is 3 and the common difference is -21 Answers with solutions a = 1 and d = a 2 - a 1 = 7 – 1 = 6 a n =a 1 +(n-1)d = 1 + (n-1) 6 = 1+6n-6 So a n = 6n-5 a 10 = 6(10) – 5 = 55 a 25 = 6(25)-5 = 145 a = x+10 and d = a 2 - a 1 = x+7-(x+10) = -3 a n =a 1 +(n-1)d = x+10 + (n-1)(-3) = x+10-3n+3 So a n = x-3n+13 a 10 = x -3(10)+13 = x - 17 a 25 = x -3(25)+13 = x - 62 a = 3 and d = -21 a n =a 1 +(n-1)d = 3 + (n-1) -21 = 3-21n+21 So a n = 24-21n a 10 = 24-21(10) = -186 a 25 = 24-21(25) = -501 a = = 5 and d = 3 a n =a 1 +(n-1)d = 5 + (n-1) 3 = 5+3n-3 So an an = 3n+2 a 10 = 3(10) +2 = 32 a 25 = 3(25)+2 = 77

Find the 14 th term of the arithmetic sequence with first term of 5 and the common difference is –6. a n = a 1 + (n – 1) d a 14 = You are looking for the term! List which variables from the general term are provided! The 14 th term in this sequence is the number -73! a = 5 and d = -6 5 (-6) = 5 + (13) * (-6) = = -73

In the arithmetic sequence 4,7,10,13,…, which term has a value of 301? a n = a 1 + (n – 1) d You are looking for n! The 100 th term in this sequence is 301!

In an arithmetic sequence, term 10 is 33 and term 22 is –3. What are the first four terms of the sequence? The sequence is 60, 57, 54, 51, ……. Use what you know! a 10 =33 a 22 = -3 a n = a 1 + (n – 1) d For term 10: 33= a 1 + 9d a n = a 1 + (n – 1) d For term 22 : -3= a d HMMM! Two equations you can solve! 33 = a 1 +9d -3 = a 1 +21d By elimination 36 = -12d -3 = d SOLVE: 33 = a 1 + 9d 33 = a 1 +9(-3) 33 = a 1 –27 60 = a 1

Vocabulary of Sequences (Universal) WRITE THIS FORMULA DOWN ON A HINTS CARD!!!

Given an arithmetic sequence with x x = 80 Ex 7:

Try this one: x 0.5 OMIT

9 x x = 27 Ex 8:

x Ex 9: OMIT

Sooooooo ………… Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term But what if we multiply? Then we get Geometric Sequences!!!

Vocabulary of Sequences (Universal) WRITE THIS FORMULA DOWN ON YOUR HINTS CARD!!!

Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic Ex 10:

1/2 x 9 2/3 Ex 11:

-3, ____, ____, ____ Ex 12: OMIT

Classwork: U1D2 Front Side WS #1-11 odd #1 16, 8, 0, -8

Homework: U1D2 Front Side WS #2-12 even #1 16, 8, 0, -8 AND have your parent read the class letter then sign & return the actual Math III acknowledgement sheet to me.