Question

For what value of k, \[k + 9,2k - 1,2k + 7\] are consecutive terms of an A.P.

Answer 1

Hint:
For the term k, \[k + 9,2k - 1,2k + 7\], given are in A.P so we can state that their differences among the consecutive terms are constant. Hence, we can equate the difference of the first and second term to the difference of second and third term.

Complete step by step solution:
As the given terms are,
 \[k + 9,2k - 1,2k + 7\]

So, calculating their corresponding differences.
 \[k + 9 - 2k + 1 = 2k - 1 - 2k - 7\]
On simplifying further, we get,
 \[
   - k + 10 = - 8 \\
   \Rightarrow k = 18 \\
 \]
Hence, the value of k for which the above given terms are in A.P is for \[k = 18\] .
And again, putting back the numbers in the terms given as k, \[k + 9,2k - 1,2k + 7\] ,the numbers are
 \[
  k + 9 = 27 \\
  2k - 1 = 35 \\
  2k + 7 = 43 \\
 \]

Hence, the numbers are 27, 35, 43.

Note:
An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.
Properties of Arithmetic Progressions
1) If the same number is added or subtracted from each term of an A.P, then the resulting terms in the sequence are also in A.P with the same common difference.
2) If each term in an A.P is divided or multiplied with the same non-zero number, then the resulting sequence is also in an A.P.