Sequences

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

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

32, 16, 8, 4, , ,

3, 6, 12, 24, , ,

4, 7, 10, 13, , ,

4, 7, 10, 13, , ,

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

Arithmetic Sequence:(i) Recursive Definition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given first term.(ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by a linear equation.

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

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

To Find The Common Difference

d = tn - t(n - 1)d is the common difference

tn is an arbitrary term in the sequence

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

To Find the nth Term In an Arithmetic Sequence

tn = a + (n - 1)d

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

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

Solution: a = 11 t51 = 11 + (51 - 1)(-6)

d = 5 - 11 t51 = 11 + (50)(-6)

= -6 t51 = 11 - 300

n = 51 t51 = -289

32, 16, 8, 4, , ,

3, 6, 12, 24, , ,

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

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

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

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

To Find The Common Ratio

r is the common ratio

tn is an arbitrary term in the sequence

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

To Find the nth Term In a Geometic Sequence

tn is the nth terma is the first termn is the "rank" of the nth term in the sequencer is the common ratio