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Chapter 1: Number Patterns 1.3: Arithmetic Sequences

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  1. Chapter 1: Number Patterns1.3: Arithmetic Sequences Essential Question: What is the symbol for summation?

  2. 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

  3. 1.3 Arithmetic Sequences • In an arithmetic sequence {un} (Recursive Form): un = un-1 + dfor 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

  4. 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-1 + 1.5 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)

  5. 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-1 + 4 with u1=-7 can also be expressed as un = -7 + (n-1) ∙ 4


  6. 1.3 Arithmetic Sequences • Example 3, continued • u1 = -7 • u2 = -7 + 4 = -3 • u3 = (-7 + 4) + 4 = -7 + 2 ∙ 4 = -7 + 8 = 1 • u4 = (-7 + 2 ∙ 4) + 4 = -7 + 3 ∙ 4 = -7 + 12 = 5 • u5 = (-7 + 3 ∙ 4) + 4 = -7 + 4 ∙ 4 = -7 + 16 = 9 • 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

  7. 1.3 Arithmetic Sequences • Example 4 • 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 = = -5 + 3n – 3 = • So what would be the 8th term in this sequence? The 14th? The 483rd? -5 + (n-1)3 3n - 8

  8. 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

  9. 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

  10. 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 u6 57 = u1 + 45 57 – 45 = u1 12 = 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 = 12 + 9n – 9 un = 9n + 3, for n ≥ 1

  11. 1.3 Arithmetic Sequences • Today’s assignment: Page 29, 1-6, 25-30 Ignore directions for graphing • Tomorrow: Summation NotationCalculators will be important

  12. Chapter 1: Number Patterns1.3: Arithmetic Sequences (Day 2) Essential Question: What is the symbol for summation?

  13. 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

  14. 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

  15. Version A • Using those formulas:

  16. Version B

  17. 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

  18. 1.3 Arithmetic Sequences seq (function, variable, start, end) sum seq(7-3x,x,1,50) = -3475

  19. 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:

  20. 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)

  21. 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