Question

If $k + 2,k,3k - 2$ are three consecutive terms of an AP then what is the value of k?
$
  {\text{A}}{\text{. 0}} \\
  {\text{B}}{\text{. 6}} \\
  {\text{C}}{\text{. 5}} \\
  {\text{D}}{\text{. 8}} \\
 $

Answer 1

Hint- Here, we will proceed by using the condition for three consecutive terms to be in AP which is $2b = a + c$ where b is the middle term, a is first term and c is the third term. From here we will get one equation in one variable i.e. k.

“Complete step-by-step answer:”
Given, $k + 2, k ,3k - 2$ are three consecutive terms of an AP.
As we know that for the three consecutive terms a, b and c to be in an AP, twice of the second term should be equal to the sum of the first and last term i.e. $2b = a + c$
For the given terms to be in an AP, twice of the middle term which is k should be equal to the sum of the first term which is (k+2) and the second term which is (3k-2).
 $
  2k = \left( {k + 2} \right) + \left( {3k - 2} \right) \\
   \Rightarrow 2k = 4k \\
   \Rightarrow 4k - 2k = 0 \\
   \Rightarrow 2k = 0 \\
   \Rightarrow k = 0 \\
 $
Therefore, the required value of k is 0.
Hence, option A is correct.

Also, the three consecutive terms of an AP are 2 , 0 ,-2.

Note- Arithmetic progression (AP) is a series in which the common difference between any two consecutive terms remains the same. For any three consecutive terms a, b and c to be in AP, the difference between the first two terms should be equal to the difference between last two terms i.e., $
  b - a = c - b \\
   \Rightarrow b + b = c + a \\
   \Rightarrow 2b = a + c \\
 $.