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Lesson 2.2 Finding the n th term

Lesson 2.2 Finding the n th term. Writing the RULE for a Linear Sequence Homework: lesson 2.2/1-8. Objectives. Use inductive reasoning to find a pattern Create a rule for finding any term/value in the sequence Use your rule to predict any term in the sequence. 200th term?. Next term? 62.

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Lesson 2.2 Finding the n th term

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  1. Lesson 2.2Finding the nth term Writing the RULE for a Linear Sequence Homework: lesson 2.2/1-8

  2. Objectives • Use inductive reasoning to find a pattern • Create a rule for finding any term/value in the sequence • Use your rule to predict any term in the sequence

  3. 200th term? Next term? 62 20, 27, 34, 41, 48, 55, . . . How do we find this 200th term? WHY? Function Rule: The rule that gives the nth term for a sequence. n = term number (location of a value in the sequence)

  4. n = 200 7n+13 => 7(200)+13 = 1413

  5. Looking at 1, 4, 7, 10, 13, 16, 19, ......., carefully helps us to make the following observation: Common Difference: As you can see, each term is found by adding 3, a common difference from the previous term

  6. Looking at 70, 62, 54, 46, 38, ... carefully helps us to make the following observation: This time, to find each term, we subtract 8, a common difference from the previous term

  7. Writing the Rule/ nth term • Common difference (n) +/- ‘something’ n = 1 2 3 4 5 6 values = 7, 2, -3, -8, -13, -18, … -5 -5 -5 -5 • -5n +/- -5(1) = -5 +12 = 7 • nth term RULE: -5n + 12 + 12 Common Difference something +/-

  8. Finding the nth Term • Find the Common Difference • CD becomes the coefficient of n • add or subtract from that product to find the sequence value +/- x • Write the RULE 6(25)-3 6n-3 147 +6 6n -3 6n - 3

  9. Use the Rule to complete the pattern What pattern do you see consistently emerging from all these rules? Common difference Are these examples of linear or nonlinear patterns?

  10. Common Difference = -5 Adjust => -5n +/- ________ + 12 -5n + 12 Function Rule: 20th term => -88

  11. Use the pattern to find the rule & the missing term +6 +6 +6 +6 RULE: 6n+ _?__ Common difference = 6 n=1  6(1)+ _?__ = 6 n=2  6(2)+ ? =12 ? = 0 RULE: 6n

  12. +2 +2 +2 +2 RULE: 2n+ _?__ Common difference = 2 n=1  2(1)+ _?__ = 7 n=2  2(2)+ ? =9 ? = 5 RULE: 2n+5

  13. -4 -4 -4 -4 RULE: -4n+ _?__ Common difference = -4 n=1  -4(1)+ _?__ = -3 n=2  -4(2)+ ? =-7 ? = +1 RULE: -4n+1

  14. Use a table to find the number of squares in the next shape in the pattern. 1 2 3 3n+2 152 5 8 11

  15. Rules that generate a sequence with a constant difference are linear functions. Ordered pairs x y

  16. Rules for sequences can be expressed using function notation. f (n) = −5n + 12 In this case, function f takes an input value n, multiplies it by −5, and adds 12 to produce an output value.

  17. IS THE PATTERN LINEAR? YES; cd=-3 NO NO YES; cd=+4

  18. Copy and complete the table Term n 1 2 3 4 5 6 7 8 Difference n – 5 -4 -3 -2 -1 0 1 2 3 +1 4n – 3 1 5 9 13 17 21 25 29 +4 -2n + 5 3 1 -1 -3 -5 -7 -9 -11 -2 3n – 2 1 4 7 10 13 16 19 21 +3 -5n + 7 2 -3 -8 -13 -18 -23 -28 -33 -5 Function Rule Coefficient

  19. Find the next term in an Arithmetic and Geometric sequence • Arithmetic Sequence • Formed by adding a fixed number to a previous term • Geometric Sequence • Formed by multiplying by a fixed number to a previous term

  20. Arithmetic sequence formula n represents the term you are calculating 1st term in the sequence d the common difference between the terms

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