Question

If \[k+2,4k-6,3k-2\] are the three consecutive terms of an A.P then the value of \[k\] is:
(a) 2
(b) 3
(c) 3
(d) 5

Answer 1

Hint: We solve this problem by using the condition of three terms of an A.P
We have the condition that is \[a,b,c\] the three terms of an A.P then
\[2b=a+c\]
Bu using the above condition to given three terms of the A.P we find the required value.

Complete step by step answer:
We are given that the three terms of an A.P as
\[k+2,4k-6,3k-2\]
Now, let us use the standard condition of an A.P
We know that the condition of an A.P that is \[a,b,c\] the three terms of an A.P then
\[2b=a+c\]
By using the above condition to given three terms of the A.P we get
\[\begin{align}
  & \Rightarrow 2\left( 4k-6 \right)=k+2+3k-2 \\
 & \Rightarrow 8k-12=4k \\
\end{align}\]
Now, let us interchange the terms in such a way that the variable terms comes one side and constant terms on the other side then we get
\[\begin{align}
  & \Rightarrow 8k-4k=12 \\
 & \Rightarrow 4k=12 \\
 & \Rightarrow k=3 \\
\end{align}\]
Therefore we can conclude that the value of \[k\] is 3.
So, option (c) is the correct answer.

Note:
We can solve this problem by using the other method.
We know that the general representation of terms of A.P as
\[a,a+d,a+2d,....\]
Where \[a\] is the first term and \[d\] is a common difference.
Here, we know that the common difference is obtained by subtracting any two consecutive terms.
We are given that the terms of A.P as
\[k+2,4k-6,3k-2\]
Let us assume that the common difference as \[d\]
By subtracting first term from second term we get the common difference as
\[\begin{align}
  & \Rightarrow d=\left( 4k-6 \right)-\left( k+2 \right) \\
 & \Rightarrow d=3k-8 \\
\end{align}\]
Now, by subtracting the second term from third term we get the common difference as
\[\begin{align}
  & \Rightarrow d=\left( 3k-2 \right)-\left( 4k-6 \right) \\
 & \Rightarrow d=-k+4 \\
\end{align}\]
We know that the common difference will be same through the A.P
By using the above statement we get
\[\begin{align}
  & \Rightarrow 3k-8=-k+4 \\
 & \Rightarrow 4k=12 \\
 & \Rightarrow k=3 \\
\end{align}\]
Therefore we can conclude that the value of \[k\] is 3.
So, option (c) is the correct answer.