High School

Identify the pattern in each sequence:

a) 2013, 9, 27, 81, 243
b) 5, 25, 125, 625, 3125
c) 128, 64, 32, 16, 8, 4, 2
d) 100000, 10000, 1000, 100, 10

Answer :

The sequences given in each part of the question seem to involve patterns about numbers, and these sequences can be interpreted based on mathematical concepts such as exponential and geometric progressions. Let's analyze each sequence one by one:

  1. For sequence (a): [tex]2013, 9, 27, 81, 243[/tex], it appears that almost all terms, except the first, follow a pattern. If we look at the numbers [tex]9, 27, 81, 243[/tex], they are powers of [tex]3[/tex]:

    • [tex]9 = 3^2[/tex]
    • [tex]27 = 3^3[/tex]
    • [tex]81 = 3^4[/tex]
    • [tex]243 = 3^5[/tex]

    It could be inferred that this sequence involves powers of the number [tex]3[/tex], starting with [tex]3^2[/tex]. The number [tex]2013[/tex] does not fit this pattern, so it may have been added erroneously or could represent a specific value based on additional context not provided here.

  2. For sequence (b): [tex]5, 25, 125, 625, 3125[/tex], these numbers represent powers of [tex]5[/tex]:

    • [tex]5 = 5^1[/tex]
    • [tex]25 = 5^2[/tex]
    • [tex]125 = 5^3[/tex]
    • [tex]625 = 5^4[/tex]
    • [tex]3125 = 5^5[/tex]

    This sequence is a geometric progression where each term is the previous term multiplied by [tex]5[/tex].

  3. For sequence (c): [tex]128, 64, 32, 16, 8, 4, 2[/tex], these numbers are descending powers of [tex]2[/tex]:

    • [tex]128 = 2^7[/tex]
    • [tex]64 = 2^6[/tex]
    • [tex]32 = 2^5[/tex]
    • [tex]16 = 2^4[/tex]
    • [tex]8 = 2^3[/tex]
    • [tex]4 = 2^2[/tex]
    • [tex]2 = 2^1[/tex]

    In this sequence, each term is half the previous term, forming a geometric series descending by a factor of [tex]2[/tex].

  4. For sequence (d): [tex]100000, 10000, 1000, 100, 10[/tex], these numbers can be seen as powers of [tex]10[/tex], stepping downwards:

    • [tex]100000 = 10^5[/tex]
    • [tex]10000 = 10^4[/tex]
    • [tex]1000 = 10^3[/tex]
    • [tex]100 = 10^2[/tex]
    • [tex]10 = 10^1[/tex]

    This sequence shows a geometric progression with ratio [tex]\frac{1}{10}[/tex] as each number is [tex]\frac{1}{10}[/tex] of the previous.

Overall, these sequences illustrate the concept of geometric sequences where each term is either a constant multiple or fraction of the previous term.

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