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Guess my rule! - 1
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The sequence 1, 4, ... might be continued in many ways. Here are four possibilities and the rules that they follow.

1, 4, 7, 10, . . . an = 3n - 2, an arithmetic sequence
1, 4, 16, 64, . . . an = 4n-1, a geometric sequence
1, 4, 27, 256, . . . an = nn
1, 4, 1, 4, . . . an = 4? (This solution has been left
unfinished. You may figure out the missing
part as you complete the problems below!)
Each number in the sequence is an, with n representing the position in the sequence. Another way to write the first formula is f(n) = 3n - 2.
For each sequence, think of at least two rules that might describe the sequence. State the rules and show how the sequence continues. Express your rules algebraically if you can.
  1. 0, 1, . . .
  2. 1, -1, . . .
  3. 2, 1 2, . . .
  4. 10, 100, . . .
  5. 10, 101, . . .
  6. 1 3, 1 2, . . .

Compare your findings with those of your classmates. Compile a class list of solutions for each item.

Hints
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Try to start out with arithmetic and geometric sequences (unless that is impossible). Then let your fantasy guide you in finding more solutions.

Answers
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See solutions.

Solutions
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Possible solutions are given for each.


  1. 0, 1, 2, 3, ... an = n - 1
    0, 1, 3, 7, ... an = 2n-1 - 1
    0, 1, 0, 1... an = 1+(-1)n ---2---
    0, 1, 12, 1... an = n+(n--11)n+1-
    0, 1, log1019, log1028, ... an =log10(9n - 8)
    0, 1, 0, -1, ... an =sin(p 2(n - 1))
    0, 1, 10, 11, ... an = n - 1 (base 2)

  2. 1, -1, -3, -5, ... an = 3 - 2n
    1, -1, 1, -1, ... an = (-1)n+1
    1, -1, -3, -7, ... an = 1 - 2n-1
    1, -1, -2.5, -3.75, ... an = -n + 22-n
    1, -1, 1, -1, ... an =cos(p(n - 1))

  3. 2, 1 2, -1, -2.5, ... an = 3.5 - 1.5n
    2, 1 2, 2 9, 1 8, ... an = 2n-2
    2, 12, 18, 132, ... an = 23-2n
    2, 1 2, 2, 1 2, ... an = 2(-1)n+1

  4. 10, 100, 190, 280, ... an = 90n - 80
    10, 100, 1000, 10000, ... an = 10n
    10, 100, 10000, 100000000, ... an = 10(2n-1)
    10, 100, 661.5, 2642.08, ... an = n6+6n- 72-n × 102-n

  5. 10, 101, 192, 283, ... an = 91n - 81
    10, 101, 1020.1, 10303.01, ... an = 10(101 10-)n-1
    an = . . . any answer from 1 plus
    any answer from 4!

  6. 1 3, 1 2, 2 3, 5 6, ... an = 1 3 + 1 6(n - 1)
    13, 12, 34, 98, ... an = 13(32)n-1
    1 3, 1 2, 3 5, 2 3, ... an = -n-- n+2
    1 3, 1 2, 1, undefined, -1, ... an = -1-- 4-n
    As long as n/=4

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