You create a linear sequence by starting with a number and then consistently adding (or subtracting) the same value to get the next term. This constant value added or subtracted is called the common difference.
Here's a breakdown of how to create a linear sequence:
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Choose a Starting Number: This will be the first term in your sequence. For example, let's choose 3.
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Determine the Common Difference: This is the number you will add (or subtract) each time. Let's choose a common difference of 5. If you subtract, you would use a negative number like -2.
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Generate the Sequence: Add the common difference to the previous term to get the next term.
- Term 1: 3
- Term 2: 3 + 5 = 8
- Term 3: 8 + 5 = 13
- Term 4: 13 + 5 = 18
- And so on...
So, the linear sequence is: 3, 8, 13, 18, ...
Formula Representation:
A linear sequence can be represented by the formula:
a_n = a_1 + (n - 1)d
Where:
a_nis the nth term of the sequence.a_1is the first term of the sequence.nis the term number (e.g., 1 for the first term, 2 for the second term, etc.).dis the common difference.
Example using the formula:
Let's find the 10th term in the sequence 3, 8, 13, 18...
a_1= 3n= 10d= 5
a_10 = 3 + (10 - 1) * 5
a_10 = 3 + (9) * 5
a_10 = 3 + 45
a_10 = 48
Therefore, the 10th term in the sequence is 48.
Key Characteristics of a Linear Sequence:
- Constant Difference: The difference between consecutive terms is always the same.
- Straight Line Graph: If you plot the terms of the sequence on a graph (with the term number on the x-axis and the term value on the y-axis), you'll get a straight line.
In summary, creating a linear sequence involves selecting a starting value and consistently adding or subtracting the same amount (the common difference) to generate subsequent terms. This can be expressed using a simple formula, allowing you to calculate any term in the sequence.
