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Sequence and Series (Arithmetic and Geometric Progressions)

Sequences

A sequence is a series of numbers increasing or decreasing by a constant difference between terms (a progression). As a math function a sequence has a domain of positive integers. The Sequence Function a1, a2, a3, … denotes a sequence. The values a1, a2, a3… are terms of the sequence (a1 the first term, a2 the second term, …, and continuing).

A sequence is increasing if an < an + 1 for all n and decreasing if an > an + 1 for all n. This states that for a sequence to increase the terms progress larger for each larger subscript n, and for decreasing sequence, terms progress smaller as each subscript gets larger.

Arithmetic Sequence

To find the sequence values of the first 4 terms defined by an = n − 4:

a1 = 1 − 4 = −3

a2 = 2 − 4 = −2

a3 = 3 − 4 = −1

a4 = 0

To find the sequence values of the first 5 terms defined by the formula an = 1 / 4n:

(1 divided by 4n)

a1 = 1 / 4(1) = 1/4

a2 = 1 / 4(2) = 1/8

a3 = 1 / 4(3) = 1/12

a4 = 1 / 4(4) = 1/16

a5 = 1 / 4(5) = 1/20

If each term after the first term can be determined by adding the same number to the preceding term the sequence is an arithmetic sequence and there exists a number, d, that is a common difference between each term of the sequence. This means that if we know two terms of a sequence we can determine the common difference, apply the common difference and discover additional terms of that sequence.

By the common difference we can write additional terms for the sequence 1, 3, 5 …

The common difference is 2. The next three terms are 7, 9 and 11.

We can write the next three terms for the arithmetic sequence 1, 5/8, 1/4 …

The common difference is negative 3/8 (1 = 8/8). The next three sequence terms
are: −1/8, −1/2 and −7/8.

Because each term after the first term of an arithmetic sequence is determined by adding the same difference number the second term is a1 + d, where d is the common difference. The third term can then be determined by a1 + 2d and can be stated as:

an = an − 1 + d  for n > 1

From this we can also see that from a1 any sequence value, an, beyond a1 can be determined by:

an = a1 + (n − 1)d  for n > 1

As an example, the 5th term of the arithmetic sequence having a first term of 3 and common difference of −2 is:

a5 = 3 + (5 − 1) (−2) = 3 − 8 = −5

The sequence is {3, 1, −1, −3, −5, …}

Geometric Sequence

If each term beyond the first term can be found by multiplying the preceding term by the same number the sequence is geometric. A geometric sequence is a series of numbers that has a common ratio, r, such that:

an = (r) an − 1 for n > 1

By applying this formula we can determine the next 3 terms of the geometric sequence 1, 3, 9 by first finding the common ratio, r:

We’ll use the sequence values a2 and a3 to determine the common ratio …

a2 = 3 = an − 1
(The sequence value prior to 9)

a3 = 9 = an (The sequence value n)

a3 = (r) an − 1 = (r) a2 = (r) 3 = 9

a3 = (r) 3 = 9 when r = 3

The common ratio, r, is 3.

We could have just as easily used a1 and a2 instead of a2 and a3 …

a1 = 1 = an − 1
(The sequence value prior to 3)

a2 = 3 = an (The sequence value n)

a2 = (r) a1 = 3

a2 = (r) 1 = 3 when r = 3

The next three terms of the sequence are 27, 81 and 243…

an = (r) (an − 1)

a4 = (3) 9 = 27

a5 = (3) 27 = 81

a6 = (3) 81 = 243

Each term following the first term is a power of r of the first term. It means that once the common ratio has been determined any nth term of a geometric sequence can be found by multiplying the value of a1 by r raised to the power of n − 1 and can be stated as the Geometric Sequence Formula:

an = a1 r n − 1

As an example of the Geometric Sequence Formula the next 5 terms of the sequence 1, 3, 9 with a common ratio r = 3 are determined as follows:

Given sequence 1, 3, 9 then a1 = 1, a2 = 3, a3 = 9 …

a4 = (1) 34 − 1 = 27

a5 = (1) 35 − 1 = 81

a6 = (1) 36 − 1 = 243

a7 = (1) 37 − 1 = 729

a8 = (1) 38 − 1 = 2,187

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