Finding the common difference and the nth term of an arithmetic sequence is straightforward. Let's break down the process:
Understanding Arithmetic Sequences
An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. This constant difference is called the common difference (d). Each term in the sequence can be found by adding the common difference to the previous term.
Finding the Common Difference (d)
To find the common difference, simply subtract any term from the term that immediately follows it.
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Formula:
d = a<sub>n+1</sub> - a<sub>n</sub>wherea<sub>n+1</sub>is any term anda<sub>n</sub>is the term before it. -
Example: In the sequence 2, 5, 8, 11..., the common difference is:
5 - 2 = 3(or8 - 5 = 3,11 - 8 = 3, etc.).
Finding the nth Term (an)
The nth term of an arithmetic sequence represents the value of the term at position 'n' in the sequence.
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Formula:
a<sub>n</sub> = a<sub>1</sub> + (n - 1)dwhere:a<sub>n</sub>is the nth term.a<sub>1</sub>is the first term of the sequence.nis the position of the term in the sequence.dis the common difference.
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Example: Let's find the 10th term (a10) of the sequence 2, 5, 8, 11...
a<sub>1</sub> = 2d = 3n = 10a<sub>10</sub> = 2 + (10 - 1) * 3 = 2 + 27 = 29
Therefore, the 10th term of this sequence is 29.
Practical Insights and Solutions
- If you are given the nth term and the common difference, you can work backward to find the first term (
a<sub>1</sub>) and then apply the nth term formula to find other terms. - Many online calculators and tools are available to help you determine the nth term and common difference of an arithmetic sequence quickly. Simply input the known terms of the sequence and let the software do the calculation.
This detailed explanation combines information gleaned from various resources, including those provided in the reference section, to comprehensively answer the question.
