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What is a Constant Difference in a Number Pattern?

Published in Number Sequences 2 mins read

A constant difference in a number pattern refers to the consistent value added or subtracted between consecutive terms in a sequence.

In simpler terms, if you subtract any term from the term that follows it, and you always get the same number, then that number is the constant difference. Such a sequence is called an arithmetic sequence.

Here's a breakdown:

  • Definition: A constant difference occurs when the difference between any two consecutive numbers in a sequence remains the same.

  • Arithmetic Sequence: A sequence with a constant difference is specifically called an arithmetic sequence or arithmetic progression.

  • Finding the Constant Difference: To find the constant difference, subtract any term from its subsequent term.

    Constant Difference (d) = a(n+1) - a(n)

    where a(n+1) is the next term and a(n) is the current term.

Examples:

  • Example 1: Arithmetic Sequence with Addition

    Consider the sequence: 2, 5, 8, 11, 14...

    • 5 - 2 = 3
    • 8 - 5 = 3
    • 11 - 8 = 3
    • 14 - 11 = 3

    The constant difference is 3. Each term is obtained by adding 3 to the previous term.

  • Example 2: Arithmetic Sequence with Subtraction

    Consider the sequence: 20, 15, 10, 5, 0...

    • 15 - 20 = -5
    • 10 - 15 = -5
    • 5 - 10 = -5
    • 0 - 5 = -5

    The constant difference is -5. Each term is obtained by subtracting 5 from the previous term.

Key Characteristics:

  • Linear Growth/Decline: Arithmetic sequences exhibit linear growth (addition) or decline (subtraction). This is because the value changes by a fixed amount each time.

  • Predictability: Because the difference is constant, you can predict future terms in the sequence.

  • Formula: The general formula for the nth term of an arithmetic sequence is:

    a(n) = a(1) + (n - 1)d

    where:

    • a(n) is the nth term
    • a(1) is the first term
    • n is the term number
    • d is the constant difference

In summary, a constant difference in a number pattern indicates a predictable and linear relationship between the terms, forming an arithmetic sequence.

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