Sequences all around us

patterns warped and otherwise by flickr user Grant MacDonald

Find the next three terms in each sequence of numbers ...

1, 1, 2, 3, 5, 8,13, , ,

3, 6, 12, 24, , ,

4, 7, 10, 13, , ,

32, 16, 8, 4, , ,

4, 7, 10, 13, , , 16 19 22

RANK

Sequence: An ordered list of numbers that follow a certain pattern (or rule).

Arithmetic Sequence:

Example:

(ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by a linear equation.

(i) Recursive Definition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given first term.

Sequence: An ordered list of numbers that follow a certain pattern (or rule).

(ii) From the implicit definition, d is the slope of the linear equation.

(i) The number that is repeatedly added to successive terms in an arithmetic sequence.

Common Difference (d):

Example: 4, 7, 10, 13, , ,

To Find The Common Difference

d is the common differencetn is an arbitrary term in the sequencet(n - 1) is the term immediately before tn in the sequence

d = tn - t(n - 1)

Example: Find the common difference for the sequence:

11, 5, -1, -7, ...

5 - 11= -6

-1 - 5 = -6

-7 - (-1) = -6

d = -6

Solution: a = 11 t51 = 11 + (51 - 1)(-6) d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289

Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ...

tn is the nth terma is the first termn is the "rank" of the nth term in the sequenced is the common difference

tn = a + (n - 1)d

To Find the nth Term In an Arithmetic Sequence

3, 6, 12, 24, , ,

3, 6, 12, 24, , ,

Geometic Sequence:

(ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by an exponential equation.

(i) Recursive Definition: An ordered list of numbers generated by continuously multiplying a value (the common ratio) with a given first term.

Common Ratio (r):

(ii) From the implicit definition, r is the base of the exponential function.

(i) The number that is repeatedly multiplied to successive terms in a geometic sequence.

To Find The Common Ratio

t(n - 1) is the term immediately before tn in the sequence

tn is an arbitrary term in the sequence

r is the common ratio

To Find the nth Term In a Geometic Sequence

r is the common ratio

n is the "rank" of the nth term in the sequence

a is the first term

tn is the nth term

32, 16, 8, 4, , ,

Write the implicit definition for this sequence.

Some "quickies" to get us started ...

Find the value(s) of r in .

In the geometric sequence, if = 3 and r = 2 , find .

If the first term of a geometric progression is and the common ratio is -3, find the next three terms.

Determine the common ratio for the geometric sequence: