Pre-Cal 40S June 3, 2009

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: