Precalculus: Arithmetic and Geometric Sequences

-An arithmetic sequence has a constant difference and can be expressed as a line.

-A geometric pattern is a not linear, it is exponential.

Arithmetic Sequence:

In an arithmetic sequence {U_n}

U_n=U_n-1+d

And for some constant d and all n>2

Example:

If {U_n} is an arithmetic sequence with U₁= 3 and U₂ = 4.5 as its first two terms,

a) a) find the common difference

b) b) write the sequence as a recursive function

c) c) give the first seven terms of the sequence

Solution:

A) a) the sequence is arithmetic and has a common difference of U₂-U₁ = 4.5 - 3 = 1.5

B) b) the recursive function that describes the sequence is U₁ = 3 and U_n-1 + 1.5 for n>2

C) c) the first seven terms are 3, 4.5, 6, 7.5, 9, 10.5 and 12

Geometric Sequence:

In a geometric sequence {U_n}

U_n=RU_n-1

For some U₁ and some nonzero constant r and all n>2

Example:

Is the following sequence geometric? If so, what is the common ratio? Write each sequence as a recursive function.

{3,9,27,81...}

The sequence is geometric with a common ratio of 3.

U₂/U₁ = 9/3 = 3

U₃/U₂=27/9 = 3

U₄/U₃ = 81/27 = 3

Because each term is obtained by multiplying the precious term by 3, the sequence may be denoted as a recursive function

Precalculus