Chapter 1: Number Patterns 1.3: Arithmetic Sequences Essential Question: What is the symbol for summation?

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

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

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)

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

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

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 = = -5 + 3n – 3 = So what would be the 8th term in this sequence? The 14th? The 483rd? -5 + (n-1)3 3n - 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

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

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

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

Chapter 1: Number Patterns 1.3: Arithmetic Sequences (Day 2) Essential Question: What is the symbol for summation?

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

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

Version A Using those formulas:

Version B

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

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

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:

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)

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