Arithmetic Sequences and Series

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This article explores Arithmetic Sequences and Series. The main objectives in this article include; learning how arithmetic sequences work, how to find the common difference of an Arithmetic sequence, deriving the arithematic sequence and series formulae. The article contains lots of examples explaining how arithmetic sequences works.

What is an Arithmetic Sequence?

A sequence is an ordered list of numbers or objects. There must be a pattern in the way these numbers or objects are organised. In mathematics this pattern is called a common difference or ratio.

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The sequence shown above is known as an Arithmetic sequence. An Arithmetic Sequence is an ordered list of numbers where the difference between the successive terms is constant. The common difference in this arithmetic sequence is 4.

The terms of a sequence are commonly denoted by a single variable, say an, where the index n indicates the nth element of the sequence.

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In an arithmetic sequence the common difference has to be constant.

Formula for the nth term of the sequence

Given that a1 is the initial (first) term of an Arithmetic Progression (sequence) and d is the common difference. The formula to find the nth term of the sequence (an) is:

an = a1 + (n – 1)d

There is another version of the formula which is useful when you know a term in the sequence and its position but do not have the full sequence. Provided that you know a term (a) of a sequence in the mth position. Then the nth term is given by;

an = am + (n – m)d

Common Difference Formula

The common difference is the difference between the successive terms. When you have been given the following sequence

11, 15, 19, 23, 27, …

Subtract the first term from the second term to find the common difference;

15 – 11 = 4

Henceforth the common difference formula is given by…

The common difference (d) = an – an-1

The general idea is if second term is a sequence is a2 and the first term in the sequence is a1, then the common difference is given by;

d = a2 – a1
The progression direction an arithmetic sequence depends on the common difference. If the common difference is: Positive, the elements (terms) will grow towards positive infinity. When the common difference is Negative, the members (terms) will grow towards negative infinity.

Arithmetic Series Formula

Arithmetic Series An Arithmetic series is the sum of of all the terms in an arithmetic sequence.

The addition of all the terms in a given arithemtic sequence is called an arithmetic series. The sum for all the terms in the sequence up to nth term can be found by using the formula:

Sum up to the nth term; Sn = n/2(a1 + an)

In this formula Sn stands for sum up to the nth term, a1 is the first term in the sequence, an is the last term in the sequence.

Arithmetic Sequences

A sequence that increases by a constant amount each time is called an arithmetric sequence The following sequences are all examples of arithmetic sequences.

3, 7, 11, 15, 19, … (because 4 is added each time)
2, 4, 6, 8, 10, … (because 2 is added each time)
2, 7, 12, 17, 22, … (because 5 is added each time)
17, 14, 11, 8, … (because 4 is added each time)
a, a + d, a + 2d, a + 3d, … (because d is added each time)

Besides the general formula explained above. An arithmetic sequence can also be represented in a recurrence relationship form shown below

Uk + 1 = Uk + n

For the most part when asked to find the nth term of an arithmetic sequence the appropriate formula to use;

an = a1 + (n – 1)d

In the formula an is the nth term to be found. a1 is the first term in the arithmetic sequence, n is the position of the nth term to be found and d stands for the common difference between each sucessive term in the sequence.

Here are some examples:

Example: Given the following Arithmetic Sequence:

1, 3, 5, 7, 9, …
  1. Find the Common Difference of the sequence
  2. State the following two terms of the sequence
Answer:

  1. a1 = 1
    a2 = 3
    common difference (d) = 3 – 1 = 2
  2. 6th term = 1 + (6 – 1)2
    = 11
    7th term = 1 + (7 – 1)2
    = 13

Explanation:

  1. The first part of the question asks to find the common difference between the arithmetic sequence;

    1, 3, 5, 7, 9, …

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    To find the common difference between sequence find two corresponding terms and use subtraction to find the difference. The formula to find the the common difference states;

    The common difference (d) = an – an-1

    In this example we shall pick the first and second term. The first term is 1 and the second term is 3 as a result;

    The common difference = 3 – 1 = 2

    [svg id=”11″]

  2. The second part of the question requires to find the following two terms of the sequence. This part of the question is a conclusion to the previous answer. The formula to find the nth term of an arithmetic sequence is given by;

    an = a1 + (n – 1)d

    The furst step is to identify the known and unknown values and the plugin the known values into the formula to find the uknown value. In the previous answer the common difference (d) was 2. We must find the first and the second term in the sequence.

    [svg id=”10″]

    6th term = 1 + (6 – 1)2
    = 11
    7th term = 1 + (7 – 1)2
    = 13

    Therefore adding the 6th and 7th terms to the sequence provides the following sequence;

    1, 3, 5, 7, 9, 11, 13, …

Example: Given the following arithmetic sequence;

14, 7, 0, -7, -14, …
  1. Find the common difference of the sequence
  2. State the following two terms in the sequence
Answer:

    Explanation:

    1. The first part of the question requires to find the common difference of the sequence;

      14, 7, 0, -7, -14, …

      Use two terms that correspond to each other to find the common difference. In this example we shall use the fourth and 5th terms to find the common difference.

      [svg id=”12″]

      The 5th term is -14 and the 4th term is -17. Thefore;

      The common difference = -14 – -7 = -7

      The common difference between each succesive term is -17.

      [svg id=”13″]

    2. The second part of the question requires to find the following two terms in the sequence. Any sequence in the formula is found by the formula;

      an = a1 + (n – 1)d

      The formula requires that the common difference is known which was found in the first part of the question.

    Example: Given the following arithmetic sequence;

    2.5, 2.25, 2.0, 1.75, 1.5, …
    1. Find the common difference of the sequence
    2. State the following two terms in the sequence
    Answer:

      Explanation:

      This example involves decimals. The same principal and steps apply.

      1. The first part of the question requires to find the common difference of the Arithmetic sequence;

        2.5, 2.25, 2.0, 1.75, 1.5, …

        In this example the second and first terms of the sequence will be used to find the common difference.

        [svg id=”14″]

        The first term is 2.5 and the first term is 2.25, therefore;

        The common difference = 2.25 – 2.5 = -0.25

        The common difference is 0.25

        [svg id=”15″]

      2. [svg id=”14″]

        The second part of the question requires to find the following two terms in the sequence. The arithmetic formula can be used to find the rest od the sequence. The first term is 2.5 and the common difference is -0.25.

      Example: Find the 10th term of the sequence 1, 3, 5, 7, 9 …

      Answer:
      Explanation:

      This example requires to find the 10th term of the sequence;

      1, 3, 5, 7, 9 …

      This sequence was taken from a previous example to explain how to find any nth term in a sequence.

      [svg id=”10″]
      [svg id=”11″]

      The common difference = 3 – 1 = 2

      The first term of the sequence is 1, the question requires to find the 10th term therefore n = 10, and the common difference is 2. Plugin the known values into the nth term formula;

      a10 = 1 + (10 – 1) × 2
      = 19

      Therefore the 10th term of the sequence is 19.

      Example: Given the following arithmetic sequence;

      1, 3, 5, 7, 9 …

      Find the general expression for the nth term of the sequence.

      Answer:

        Explanation:

        The question requires to find the formula that can be used to find any nth term in the sequence. The nth term arithmetic formula will be appropriate here;

        an = a1 + (n – 1)d

        To find the nth term the formula requires that the first term is known a1, the position (n) of the term to be found is known and the common difference of the sequence is known. Using the previous example answer the common difference was found to be 2. Note here that a general expression to find any nth term in the sequence is required. Therefore the n value is not required.

        an = 1 + 2(n – 1)
        = 1 + 2n – 2
        an = 2n -1

        Arithmetic Series

        An Arithmetic Series is the sum of a finite Arithmetic sequence.. For example consider the arithmetic sequence

        1, 3, 5, 7, 9

        The arithmetic Series for this arithmetic sequence iis rewritten as;

        1 + 3 + 5 + 7 + 9

        The quickest way to find the sum of this artithmetic sequence is to count the number of terms. In this example the number of terms in the sequence is 5.The multiply the number of terms by the sum of the first and last number in the sequence. In this example the first term is 1 and the last term is 9.

        1 + 9 = 10

        Then divide the answer by 2;

        1 + 3 + 5 + 7 + 9 = [ 5 ( 1 + 9 ) ] / 2 = 25

        This formula works for any real numbers a1 and an and can be summarised with this formula:

        Sum of nth terms Sn = n/2(a1 + an)

        Example: Find the last term and the sum of the following sequence up to 26 terms

        2 + 6 + 10 + 14 + …
        Answer:

        Last term = a + (n – 1)d = 2 + 25 × 4 = 102
        Sum to 26 terms = 26/2(2×2 + (26 – 1) × 4)
        = 13(4 + 25×4)
        = 1352

        Explanation:
        EXPLANATION REQUIRED…

        Example: Find the sum of the first 1000 natural numbers.

        Answer:

        1, 2, 3, 4, …
        = n/2(2a + (n – 1)×d)
        = n/2(2×1 + (n -1)×1)
        = 1000/2(1000 + 1)

        Explanation:

        The natural numbers are numbers that count from 1 and increase with a common difference of 1.

        1, 2, 3, 4, …

        The first term is 1 and the common difference of the sequence is 1. The question requires to find the find the sum up to the 1000 term. First identify the formula to use. The sequence is an arithmetic sequence since the common difference is constant.

        Sum up to the nth term; Sn = n/2(a1 + an)

        The next step is to identify the known and unknown values to plugin into the formula to find the unknown value. In this example n = 1000, d = 1 and a1 = 1. Therefore;

        = n/2(2a + (n – 1)×d)

        [EXPLANATION INCOMPLETE]

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