Question

# For what values of k will ${\text{k + 9, 2k - 1 and 2k + 7 }}$ are consecutive terms of an A.P.?

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Hint: Use the properties of Arithmetic Progression. i.e. The difference between any two consecutive terms of an arithmetic progression is constant.

In the given question, it is given the three consecutive terms of an arithmetic progression in terms of a variable k.
For an arithmetic progression, we know the relation between two consecutive terms.
It is given as:
The difference between any two consecutive terms of an arithmetic progression is a constant value.
So, using this relation we will first find the difference between first term and second term and then find the difference between second term and third term.
So, on finding the difference between first term and second term, we get
$(2{\text{k - 1) - (k + 9) = 2k - 1 - k - 9 = k - 10}}$ . (1)
Now, we will find the difference between second term and third term, which is given as:
$(2{\text{k + 7) - (2k - 1) = 2k + 7 - 2k + 1 = 8}}$ (2)

Since the given three terms of the A.P. are consecutive terms. Therefore:
Second term – first term = Third term – Second term.
So, putting the values from (1) and (2), we get:
${\text{k - 10 = 8}} \\ \Rightarrow {\text{k = 18}} \\$
Therefore, for k=18, the given three consecutive numbers are in arithmetic progression.

Note: In this type of question where three consecutive terms of an A.P. are given in terms of an unknown parameter then for finding the value of the unknown parameter, we will use the relation between the consecutive terms of an arithmetic progression. The difference between any two consecutive terms of an A.P. is constant.