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Sequences finding a rule

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Sequences finding a rule

1. 2. SEQUENCES
2. 3. <ul><li>A sequence is a set of terms, in a definite order, where the terms are obtained by some rule. </li></ul>A finite sequence ends after a certain number of terms. An infinite sequence is one that continues indefinitely.
3. 4. <ul><li>For example: </li></ul>1, 3, 5, 7, … (This is a sequence of odd numbers) 1st term = 2 x 1 – 1 = 1 2nd term = 2 x 2 – 1 = 3 3rd term = 2 x 3 – 1 = 5 nth term = 2 x n – 1 = 2 n - 1 . . . . . . + 2 + 2
4. 5. NOTATION <ul><li>1st term = u </li></ul>2nd term = u 3rd term = u n th term = u . . . . . . 1 2 3 n
5. 6. OR <ul><li>1st term = u </li></ul>2nd term = u 3rd term = u n th term = u . . . . . . 0 1 2 n-1
6. 7. FINDING THE FORMULA FOR THE TERMS OF A SEQUENCE
7. 8. <ul><li>A recurrence relation defines the first term(s) in the sequence and the relation between successive terms. </li></ul>
8. 9. <ul><li>u = 5 </li></ul>u = u +3 = 8 u = u +3 = 11 u = u +3 = 3 n + 2 . . . 1 2 3 n+1 For example: 5, 8, 11, 14, … 1 2 n
9. 10. <ul><li>What to look for </li></ul><ul><li>when looking for the rule </li></ul><ul><li>defining a sequence </li></ul>
10. 11. <ul><li>Constant difference: coefficient of n is the difference </li></ul><ul><li>2 nd level difference: compare with square numbers </li></ul><ul><li>( n = 1, 4, 9, 16, …) </li></ul><ul><li>3 rd level difference: compare with cube numbers </li></ul><ul><li>( n = 1, 8, 27, 64, …) </li></ul><ul><li>None of these helpful: look for powers of numbers </li></ul><ul><li>(2 = 1, 2, 4, 8, …) </li></ul><ul><li>Signs alternate: use (-1) and (-1) </li></ul><ul><li>-1 when k is odd +1 when k is even </li></ul>k k 2 3 n - 1
11. 12. <ul><li>EXAMPLE: </li></ul><ul><li>Find the next three terms in the sequence 5, 8, 11, 14, … </li></ul>
12. 13. <ul><li>EXAMPLE: </li></ul><ul><li>The nth term of a sequence is given by x = </li></ul><ul><li>Find the first four terms of the sequence. </li></ul><ul><li>b) Which term in the sequence is ? </li></ul><ul><li>c) Express the sequence as a recurrence relation. </li></ul>1 __ 2 n n 1 1024 ____
13. 14. <ul><li>EXAMPLE: </li></ul><ul><li>Find the nth term of the sequence +1, -4, +9, -16, +25, … </li></ul>
14. 15. <ul><li>EXAMPLE: </li></ul><ul><li>A sequence is defined by a recurrence relation of the form: </li></ul><ul><li>M = aM + b . </li></ul><ul><li>Given that M = 10, M = 20, M = 24, find the value of a and the </li></ul><ul><li>value of b and hence find M . </li></ul>n + 1 1 3 2 4