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**Chapter 1: Number Patterns 1.3: Arithmetic Sequences**

Essential Question: What is the symbol for summation?

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1.3 Arithmetic Sequences Arithmetic Sequence → a sequence where the difference between each term and the preceding term is constant (adding/subtracting the same number repeatedly) Example 1: Arithmetic Sequence Are the following sequence arithmetic? If so, what is the common difference? { 14, 10, 6, 2, -2, -6, -10 } { 3, 5, 8, 12, 17 } Yes; common difference is -4 Not arithmetic

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1.3 Arithmetic Sequences In an arithmetic sequence {un} (Recursive Form): un = un-1 + d for some constant d and all n ≥ 2 Example 2: Graph of an Arithmetic Sequence If {un} is an arithmetic sequence with u1=3 and u2=4.5 as the first two terms Find the common difference Write the recursive function Give the first seven terms of the sequence Graph the sequence

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**1.3 Arithmetic Sequences Example 2: Graph of an Arithmetic Sequence**

If {un} is an arithmetic sequence with u1=3 and u2=4.5 as the first two terms Find the common difference Write the recursive function Give the first seven terms of the sequence Graph the sequence d = 4.5 – 3 = 1.5 u1 = 3 and un = un for n ≥ 2 3, 4.5, 6, 7.5, 9, 10.5, 12 See page 22, figure 1.3-1b. Note that the graph is a straight line (this will become relevant later)

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**1.3 Arithmetic Sequences Explicit Form of an Arithmetic Sequence**

Arithmetic sequences can also be expressed in a form in which a term of the sequence can be found based on its position in the sequence Example 3: Explicit Form of an Arithmetic Sequence Confirm that the sequence un = un with u1=-7 can also be expressed as un = -7 + (n-1) ∙ 4

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**1.3 Arithmetic Sequences Example 3, continued**

Explicit Form of an Arithmetic Sequence un = u1 + (n-1)d for every n ≥ 1 u1 → 0 “4”s u2 → 1 “4” u3 → 2 “4”s u4 → 3 “4”s u5 → 4 “4”s

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**1.3 Arithmetic Sequences Example 4 Solution**

Find the nth term of an arithmetic sequence with first term -5 and common difference 3 Solution Because u1 = -5 and d = 3, the formula would be: un = u1 + (n-1)d un = = n – = So what would be the 8th term in this sequence? The 14th? The 483rd? -5 + (n-1)3 3n - 8

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**1.3 Arithmetic Sequences Formula: un = u1 + (n – 1)(d)**

Example 5: Finding a Term of an Arithmetic Sequence What is the 45th term of the arithmetic sequence whose first three terms are 5, 9, and 13? Solution u1 = 5 The common difference, d, is 9-5 = 4 u45 = u1 + (45 – 1)d = 5 + (44)(4) = 181

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1.3 Arithmetic Sequences Example 6: Finding Explicit and Recursive Formulas If {un} is an arithmetic sequence with u6=57 and u10=93, find u1, a recursive formula, and an explicit formula for un Solution: The sequence can be written as: …57, ---, ---, ---, 93, ... u6, u7, u8, u9, u10 The common difference, d, can be found like the slope

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1.3 Arithmetic Sequences The value of u1 can be found using the explicit form of an arithmetic sequence, working backwards: un = u1 + (n-1)d u6 = u1 + (6-1)(9) We don’t know u1 , but we know u = u – 45 = u = u1 Now that we know u1 and d, the recursive form is given by: u1 = 12 and un = un-1 + 9, for n ≥ 2 The explicit form of the arithmetic sequence is given by: un = 12 + (n-1)9 un = n – 9 un = 9n + 3, for n ≥ 1

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1.3 Arithmetic Sequences Today’s assignment: Page 29, 1-6, Ignore directions for graphing Tomorrow: Summation Notation Calculators will be important

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**Chapter 1: Number Patterns 1.3: Arithmetic Sequences (Day 2)**

Essential Question: What is the symbol for summation?

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**1.3 Arithmetic Sequences Summation Notation**

When we want to find the sum of terms in a sequence, we use the Greek letter sigma: ∑ Example 7

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**1.3 Arithmetic Sequences Computing sums with a formula**

Either of the following formulas will work: What they mean: Add the first & last values of the sequence, multiply by the number of times the sequence is run, divide by 2 Number of times sequence is run (k) multiplied by 1st value. Add that to k(k-1), divided by 2, multiplied by the common difference

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Version A Using those formulas:

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Version B

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**1.3 Arithmetic Sequences Computing sums with the TI-86**

Storing regularly used functions: 2nd, Catlg-Vars (Custom), F1, F3 Move down to function desired, store with F1-F5 We will need sum & seq( for this section Using sequence: seq(function, variable, start, end) Using sum: sum sequence

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1.3 Arithmetic Sequences seq (function, variable, start, end) sum seq(7-3x,x,1,50) = -3475

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**1.3 Arithmetic Sequences Partial Sums**

The sum of the first k terms of a sequence is called the kth partial sum Example 9: Find the 12th partial sum of the arithmetic sequence: -8, -3, 2, 7, … Formula method:

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1.3 Arithmetic Sequences Example 9: Find the 12th partial sum of the arithmetic sequence: -8, -3, 2, 7, … Calculator method: Need to determine equation: un=-8+(n-1)(5) [you can simplify] -8+5n-5=5n-13 sum seq(5x-13,x,1,12)

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**Assignment Page: 29 – 30 Problems: 7 – 17 & 31 – 43 (odd)**

Write the problem down Make sure to show either The formula you used when solving the sum; or What you put into the calculator to find the answer

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Geometric and arithmetic sequences

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