If we are given $\left( k-3 \right),\left( 2k+1 \right)\And \left( 4k+3 \right)$ three consecutive terms of an A.P. Find the value of k?

Hint: There is a relation between three terms say a, b, and c which is in A.P. is $2b=a+c$ so compare the three terms given in the question with a, b and c we get “a” as $‘k’ – 3$ and “b” as $2k + 1$ and “c” as $4k + 3$ then substitute these values in the relationship that we have just shown between a, b and c. Solve the equation and you will get the value of k.

Complete step-by-step solution:
In the above problem, we have given the three terms which are in A.P. as:
$\left( k-3 \right),\left( 2k+1 \right)\And \left( 4k+3 \right)$
We know that, if three terms say a, b and c are in A.P. then the relation between these three terms as:
$2b=a+c$……….. Eq. (1)
Now, on comparing a, b, c with the terms given in the above problem $\left( k-3 \right),\left( 2k+1 \right)\And \left( 4k+3 \right)$ the value of a, b and c are as follows:
$\begin{align}
  & a=k-3 \\
 & b=2k+1 \\
 & c=4k+3 \\
\end{align}$
Substituting the above values of a, b and c in eq. (1) we get,
$\begin{align}
  & 2b=a+c \\
 & \Rightarrow 2\left( 2k+1 \right)=k-3+4k+3 \\
\end{align}$
Multiplying 2 by $2k + 1$ on the left hand side of the above equation we get,
$4k+2=5k$
Subtracting $4k$ on both the sides of the above equation we get,
$2=k$
From the above solution, we have got the value of k as 2.
Hence, the value of k is 2.

Note: We can cross check whether the value of k that we are getting is correct or not by substituting this value of k in $\left( k-3 \right),\left( 2k+1 \right)\And \left( 4k+3 \right)$.
Substituting the value of k as 2 in $\left( k-3 \right),\left( 2k+1 \right)\And \left( 4k+3 \right)$ we get,
$\begin{align}
  & \left( 2-3 \right),\left( 2\left( 2 \right)+1 \right),\left( 4\left( 2 \right)+3 \right) \\
 & =-1,5,11 \\
\end{align}$
The above three terms should satisfy the condition of three terms in A.P. which is $2b=a+c$ where a is -1, b is 5 and c is 11 substituting these values in the relation between a, b and c we get,
$\begin{align}
  & 2\left( 5 \right)=-1+11 \\
 & \Rightarrow 10=10 \\
\end{align}$
As you can see that L.H.S = R.H.S so the value of k that we have solved above is correct.