Arithmetic Sequences AGENDA BELL OPENER DEBRIEF ARITHMETIC SEQUENCES
Arithmetic Sequences
AGENDA BELL OPENER & DEBRIEF ARITHMETIC SEQUENCES TERMS, EXPLICIT FORMULA, RECURSIVE FORMULA, SEQUENCES Essential Question: How does arithmetic sequence relate to linear functions?
Bell Opener. Do Now. Individual Work 1. Provide two ordered pairs that would satisfy the following function: Y = -5 X + 8 2. Find the function that would satisfy the following ordered pairs and provide a sketch of the graph. (1, 1) (4, 10) (10, 28)
Bell Opener Debrief ORDERED PAIRS IN TABLE FORMAT X 1 2 3 4 5 . . . 10 Y 1 4 7 10 13 …. 28
(1, 1) Y Intercept (0, -2)
Bell Opener Debrief Rewritten as a sequence 1 4 7 10 13 16 19 22 25 28 SEQUENCE: A STRING OF OBJECTS OR NUMBERS THAT FOLLOW A PARTICULAR PATTERN TERM: INDIVIDUAL ELEMENTS IN THE SEQUENCE
Represent the sequence using formulas EXPLICIT FORMULA RECURSIVE FORMULA Find any term in the sequence without knowing any other terms of the sequence Formula that allows you to find the nth term of the sequence if you know the (n-1)th term f(1) = 1 f(2) = 1 + 3 f(2) = f(1) + 3 f(3) = 1+ 3 x 2 f(3) = f(2) + 3 f(4) = 1+ 3 x 3 f(4) = f(3) + 3 f(5) = 1 + 3 x 4 Can you write the generalized formulas?
EXPLICIT FORMULA RECURSIVE FORMULA f(n) = 1 + 3 (n - 1) f(n) = f(n-1) + 3 Find the nth term if I know the common difference and first term Find the nth term if I know previous term Common Difference: The common term that is subtracted or added to successive terms in a sequence
Bull Dogs Growth … Can you spot the pattern? Step 5 ? Step 100? Can you Generalize the number pattern? Step 2 Step 3 Step 4
Different ways to represent the sequence 1. Explicit Formula 2. Recursive formula 3. Table 4. Graph
How many different ways can we represent the number patterns? Number Pattern (Sequence) 1, 3, 5, 7, ……. Step # of Pugs 1 1 2 3 3 5 4 7
Arithmetic Sequence Exercise (Group Work) For the following patterns, represent the sequence in four different methods. Write the sequence of numbers. Find the terms n = 5 and n = 100 Recursive Formula Explicit Formula Table Graph
Pattern 1 At one minute At two minutes At three minutes
Pattern 2 N=1 N=2 N=3
Pattern 3 What is the pattern for Area and Perimeter? N=1 N=2 N=3 Assume each edge is of 1 unit length N=4
Pattern 4 A gardener buys a plant that is 12 cm in height. Each week after, the plant grows 10 cm. What will be the height of the plant at the beginning of the 5 th and 100 th week if it follows the same pattern? Pattern 5 First three terms of a sequence: -5/2 -1 ½
Pattern 6 The following is an arithmetic sequence. 1. What is the constant difference? Describe how you found it. n f(n) 1 2 11 3 4 5 6 7 23 2. If f(0) exists, what would it be? 3. Find the explicit and recursive formula for the nth term of the sequence
Pattern 7 Suppose you have an arithmetic sequence with terms growing infinitely in both negative and positive directions. ……. f(-3) f(-2) f(-1) f(0) f(1) f(2) f(3). . . . Suppose that f(-6) = 1 and f(18) = 17 What is the constant difference? What is f(0)? What is the explicit formula for f(n)?
Back to Bell Opener 2. Find the function that would satisfy the following ordered pairs and provide a sketch of the graph. (1, 1) (4, 10) (10, 28) Can you show the sequence of number patterns? What is the common difference? What is f(0)?
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