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Determine whether a Sequence is an AP or not
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Determine whether a Sequence is an A.P. or not
Arithmetic Progression: An arithmetic progression or arithmetic sequence, is a sequence of numbers such that the differences between two successive terms is always a constant .
For example, the sequence 1,5, 9, 13, 17, 21... is an arithmetic progression with common difference 4.

STEP I Obtain an
STEP II Replace n by (n + l) in an to get an + 1
STEP III Calculate an + 1- an
STEP IV Check the value of an+ 1- an. lf an+ 1- an is independent of n, then the given sequence is an A.P. Otherwise it is not an A.P. Following examples will illustrate the above algorithm.
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Example 1: Show that the sequence defined by an = 4n + 5 is an A.P. Also, find its common difference.
Solution: you have,
         an = 4n + 5.
Replacing n by (n + 1), we get
         an + 1 = 4(n + 1) + 5 = 4n + 9
Now, an + 1 - an = (4n + 9) - (4n + 5) = 4
Clearly, an + 1 - an is independent of n and is equal to 4. So, the given sequence is an A.P. with common difference 4.

Example 2: The nth term of a sequence is 3n - 2. Is the sequence an A.P. ? If so, find its 10th term.
Solution: you have an = 3n - 2. Clearly an is a linear expression in n. So, the given sequence is an A.P. with common difference 3.
Putting n = 10,
you get a10 = 3 * 10 - 2 = 28.
so the 10th term will be 28.
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