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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? a){ 14, 10, 6, 2, -2, -6, -10 } b){ 3, 5, 8, 12, 17 } Yes; common difference is -4 Not arithmetic

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

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1.3 Arithmetic Sequences Example 2: Graph of an Arithmetic Sequence ▫If {u n } is an arithmetic sequence with u 1 =3 and u 2 =4.5 as the first two terms a)Find the common difference b)Write the recursive function c)Give the first seven terms of the sequence d)Graph the sequence d = 4.5 – 3 = 1.5 u 1 = 3 and u n = u n 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 u n = u n with u 1 =-7 can also be expressed as u n = -7 + (n-1) ∙ 4

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1.3 Arithmetic Sequences Example 3, continued ▫u 1 = -7 ▫u 2 = = -3 ▫u 3 = (-7 + 4) + 4 = ∙ 4 = = 1 ▫u 4 = ( ∙ 4) + 4 = ∙ 4 = = 5 ▫u 5 = ( ∙ 4) + 4 = ∙ 4 = = 9 Explicit Form of an Arithmetic Sequence ▫u n = u 1 + (n-1)d for every n ≥ 1 u 1 → 0 “4”s u 2 → 1 “4” u 3 → 2 “4”s u 4 → 3 “4”s u 5 → 4 “4”s

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1.3 Arithmetic Sequences Example 4 ▫Find the n th term of an arithmetic sequence with first term -5 and common difference 3 Solution ▫Because u 1 = -5 and d = 3, the formula would be: u n = u 1 + (n-1)d u n = = n – 3 = So what would be the 8 th term in this sequence? The 14 th ? The 483 rd ? -5 + (n-1)3 3n - 8

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

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1.3 Arithmetic Sequences Example 6: Finding Explicit and Recursive Formulas ▫If {u n } is an arithmetic sequence with u 6 =57 and u 10 =93, find u 1, a recursive formula, and an explicit formula for u n Solution: ▫The sequence can be written as: …57, ---, ---, ---, 93,... u 6, u 7, u 8, u 9, u 10 ▫The common difference, d, can be found like the slope

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1.3 Arithmetic Sequences The value of u 1 can be found using the explicit form of an arithmetic sequence, working backwards: u n = u 1 + (n-1)d u 6 = u 1 + (6-1)(9)We don’t know u 1, but we know u 6 57 = u – 45 = u 1 12 = u 1 Now that we know u 1 and d, the recursive form is given by: u 1 = 12 and u n = u n-1 + 9, for n ≥ 2 The explicit form of the arithmetic sequence is given by: u n = 12 + (n-1)9 u n = n – 9 u n = 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: a)Add the first & last values of the sequence, multiply by the number of times the sequence is run, divide by 2 b)Number of times sequence is run (k) multiplied by 1 st 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: 2 nd, C ATLG -V ARS (C USTOM ), 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 k th partial sum Example 9: Find the 12 th partial sum of the arithmetic sequence: -8, -3, 2, 7, … ▫Formula method:

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1.3 Arithmetic Sequences Example 9: Find the 12 th partial sum of the arithmetic sequence: -8, -3, 2, 7, … ▫Calculator method: Need to determine equation: u n =-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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