Arithmetic sequences

Arithmetic sequences

• An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.

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• The constant difference in all pairs of consecutive numbers in a sequence is called common difference, denoted by the letter “d“.

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• If the common difference between consecutive terms is positive, we say that the sequence is increasing.
• On the other hand, when the difference is negative we say that the sequence is decreasing.

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• Illustrative Examples of Increasing and Decreasing Arithmetic Sequences

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• Here are two examples of arithmetic sequences. Observe their common differences.

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• With this basic idea in mind, you can now solve basic arithmetic sequence problems.

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• Examples of How to Apply the Concept of Arithmetic Sequence

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• Example 1: Find the next term in the sequence below.

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• First, find the common difference of each pair of consecutive numbers.

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• Since the common difference is 8 or written as d = 8, we can find the next term after 31 by adding 8 to it. Therefore, we have 31 + 8 = 39.

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• Example 2: Find the next term in the sequence below.

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• Sometimes you may encounter a problem in an arithmetic sequence that involves fractions. So be ready to use your previous knowledge on how to add or subtract fractions.

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• Also, always make sure that you understand what the question is asking so that you can have the correct strategy to approach the problem.

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• In this example, we are asked to find the seventh term, not simply the next term. It is a good practice to write all the terms in the sequence and label them, if possible.

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• Now we have a clear understanding on how to work this out. Find the common difference, and use this to find the seventh term.

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• Finding the common difference,
• 3, 8, 13, 18, 23, 28, 33, 38, ...

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• Then we find the 7th term by adding the common difference of starting with the 4th term, and so on. Here’s the complete calculation.

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• Therefore, the seventh term of the sequence is zero (0).

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• For a general arithmetic sequence with first term a and a common difference d, we can generate the following terms: T1T2T3T4⋮Tn=a=T1+d=a+d=T2+d=(a+d)+d=a+2d=T3+d=(a+2d)+d=a+3d⋮⋮⋮=Tn−1+d=(a+(n−2)d)+d=a+(n−1)d

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• Therefore, the general formula for the nth term of an arithmetic sequence is: Tn=a+(n−1)d

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• Quadratic sequence A quadratic sequence is a sequence of numbers in which the second difference between any two consecutive terms is constant.

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• The general formula for the nth term of a quadratic sequence is: Tn=an2+bn+c

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