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  • Sequences and series

    Sequences and series

    Sequences

    Consider the following patter of numbers:

    2, 4, 6, 8, 10, ………

    1, 4, 9, 16, 25, 36,…….

    1, 3, 5, 7, 9, 11, 13, 15……..

    Can you be able to get the next number? Can you write the next 20 numbers from the pattern?

    There are some rules you can use to obtain the numbers in each pattern. For example in the first pattern, you can count position of the number and then multiply by 2, in other words, the nth number is simply 2n. Like 7th number is simply 2*7=14.

    The second pattern is a square of the position in the number, like 6th is 36=62, and hence next number after 36 is 72=49.

    A sequence or a progression is a series of numbers where each of the next number in the pattern can be obtained using some rules.

    Each of the numbers in the sequence is called a term.

    The rule used in the pattern ensures numbers are arranged in a definite order and the next number can always be obtained using the rule

    Example

    For each of the following sequences, find the next three terms.

    • 1, 3,5,7……..
    • 1/3, 1/9, 1/27……
    • 1,8,27……
    • -3, 6, -12……

    Solution

    • To obtain the next term, and 2 to the preceding term, hence the next term is 7+2, next 9+2, next 11+2,etc. Hence the next three terms are 9, 11, and 13.
    • The next term in the pattern is obtained by multiplying the preceding number by 1/3. That is, next number after 1/27 will be (1/27)*1/3=1/81, then next ones (1/81)*1/3=1/243 and next (1/243)*1/3=1/729.
    • In the next sequence, the next number is obtained by cubing it’s position. For instance, second number is 23 =8 and 33=27. So the next one is 43 and so on, hence the next three numbers are 64, 125 and 216.
    • The next number is obtained by multiplying the preceding term by -2. Hence the next three terms are 24,-48 and 96.
    Example

    If the nth term of a sequence is given by 2n+3, find the first, fifth and the twelfth term.

    Solution

    For the first term n=1. Hence substitute 1 in the above equation such that we have 2 x 1 + 3=5

    For the fifth term; n=5, hence second term is given by 2 x 5+3=13

    12th term is 2 x 12+3=27;

    hence the first, fifth and the twelfth term is 5, 13 and 27 consequently.

    Practice questions

    For each sequence below, give the next four terms and the rule used

    1. 1, 3, 5, 7, ……
    2. 5, 10, 15, 5, 20, …..
    3. 15, 6, -3, -12, …….
    4. 8, 4, 2,…..
    5. 1, 4, 9, 16, …..
    6. 1, 7, 49, 343,…..
    7. 1, 1/3, 1/9, 1/27, …..
    8. 230, 23, 2.3, …..
    9. 1, -1/2, 1/4, -1/8, …..
    10. 1, 1/4, 1/9, 1/16, …..
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