9-1 Mathematical Patterns

Quick Review

A sequence is an ordered list of numbers called terms.

A recursive definition gives the first term and defines the other terms by relating each term after the first term to the one before it.

An explicit formula expresses the nth term in a sequence in terms of n, where n is a positive integer.

Example

A sequence has an explicit formula a n = n 2 . eh sub n , equals , n squared , .  What are the first three terms of this sequence?

a 1 = ( 1 ) 2 = 1 Substitute  1  for  n  and evaluate . a 2 = ( 2 ) 2 = 4 Substitute  2  for  n  and evaluate . a 3 = ( 3 ) 2 = 9 Substitute  3  for  n  and evaluate . table with 3 rows and 3 columns , row1 column 1 , eh sub 1 , column 2 equals . open 1 close squared . equals 1 , column 3 cap substitute . 1 , for , n . andevaluate . . , row2 column 1 , eh sub 2 , column 2 equals . open 2 close squared . equals 4 , column 3 cap substitute . 2 , for , n . andevaluate . . , row3 column 1 , eh sub 3 , column 2 equals . open 3 close squared . equals 9 , column 3 cap substitute . 3 , for , n . andevaluate . . , end table

The first three terms are 1, 4, and 9.

Exercises

Find the first five terms of each sequence.

6. a n = − 2 n + 3 eh sub n , equals negative 2 n plus 3
7. a n = − n 2 + 2 n eh sub n , equals negative , n squared , plus 2 n
8. a n = 2 a n − 1 − 1 , eh sub n , equals . 2 . eh sub n minus 1 end sub . minus . 1 comma  where a 1 = 2 eh sub 1 , equals 2
9. a n = 1 2 a n − 1 , eh sub n , equals , 1 half . eh sub n minus 1 end sub . comma  where a 1 = 20 eh sub 1 , equals 20

Write a recursive definition for each sequence.

10. 5, 22, 39, 56, …
11. − 2 , 7 , 16 , 25 , … negative 2 comma 7 comma 16 comma 25 comma dot dot dot

Write an explicit formula for each sequence.

12. 1, 4, 7, 10, …
13. 4 , 1.5 , − 1 , − 3.5 , … 4 comma 1.5 comma , negative 1 comma , minus 3.5 comma dot dot dot

9-2 Arithmetic Sequences

Quick Review

In an arithmetic sequence, the difference between consecutive terms is constant. This difference is the common difference.

For an arithmetic sequence, a is the first term, a n eh sub n  is the nth term, n is the number of the term, and d is the common difference.

An explicit formula is a n = a + ( n − 1 ) d . eh sub n , equals , eh plus open n minus 1 close d .

A recursive formula is a n = a n − 1 + d , eh sub n , equals . eh sub n minus 1 end sub . plus . d comma  with a 1 = a . eh sub 1 , equals eh .  The arithmetic mean of two numbers x and y is the average of the two numbers x + y 2 . fraction x plus y , over 2 end fraction . .

Example

What is the missing term of the arithmetic sequence 11 , □ , 27 , … ? 11 comma white square comma 27 comma dot dot dot question mark

arithmetic mean = 11 + 27 2 = 38 2 = 19 equals . fraction 11 plus 27 , over 2 end fraction . equals , 38 over 2 , equals 19

The missing term is 19.

Exercises

Determine whether each sequence is arithmetic. If so, identify the common difference and find the 32nd term of the sequence.

14. 2, 4, 7, 10, …
15. 3, 18, 33, 48, …
16. 7, 10, 13, 16, …
17. 2, 5, 9, 14, …

Find the missing term(s) of each arithmetic sequence.

18. 1 , □ , 9 , … 1 comma white square comma 9 comma dot dot dot
19. 104 , □ , 99 , … 104 comma white square comma 99 comma dot dot dot
20. − 1 , □ , 11 , … negative 1 comma white square comma 11 comma dot dot dot
21. − 4 . 6 , □ , − 5 . 2 , … negative 4 . 6 comma white square comma negative 5 . 2 comma dot dot dot
22. − 13 , □ , □ , □ , − 3 , … negative 13 comma white square comma white square comma white square comma , negative 3 comma dot dot dot
23. 2 , □ , □ , □ , − 0 . 4 , … 2 comma white square comma white square comma white square comma negative 0 . 4 comma dot dot dot

Write an explicit formula for each arithmetic sequence.

24. − 2 , 7 , 16 , 25 , … negative 2 comma 7 comma 16 comma 25 comma dot dot dot
25. 62 , 59 , 56 , 53 , … 62 comma 59 comma 56 comma 53 comma dot dot dot

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