Question

If \[x + 1\], \[3x\], and \[4x + 2\] are in A.P., then find the value of \[x\].

Answer 1

Hint: Here, we need to find the value of \[x\]. The difference between any two consecutive terms of an A.P. is always equal. We will find the difference in the first and second term, and the difference in the second term and third term. Then, we will equate these to form a linear equation in terms of \[x\]. Finally, we will solve this linear equation and hence, find the value of \[x\].

Complete step-by-step answer:
It is given that \[x + 1\], \[3x\], and \[4x + 2\] are in A.P.
First, we will find the difference in the first and the second term.
Second term \[ - \] First term \[ = 3x - \left( {x + 1} \right)\]
Rewriting the expression, we get
\[ \Rightarrow \] Second term \[ - \] First term \[ = 3x - x - 1\]
Subtracting the like terms, we get
\[ \Rightarrow \] Second term \[ - \] First term \[ = 2x - 1\]
Now, we will find the difference in the second and the third term.
Third term \[ - \] Second term \[ = 4x + 2 - 3x\]
Subtracting the like terms, we get
\[ \Rightarrow \] Third term \[ - \] Second term \[ = x + 2\]
We know that the difference between any two consecutive terms of an A.P. is always equal.
Therefore, we get
Second term \[ - \] First term \[ = \] Third term \[ - \] Second term
Thus, we get the linear equation
\[ \Rightarrow 2x - 1 = x + 2\]
This is a linear equation in terms of \[x\]. We will solve this equation to find the value of \[x\].
Adding 1 on both sides of the equation, we get
\[\begin{array}{l} \Rightarrow 2x - 1 + 1 = x + 2 + 1\\ \Rightarrow 2x = x + 3\end{array}\]
Subtracting \[x\] from both sides of the equation, we get
\[\begin{array}{l} \Rightarrow 2x - x = x + 3 - x\\ \Rightarrow x = 3\end{array}\]
Therefore, we get the value of \[x\] as 3.

Note: An arithmetic progression is a series of numbers in which each successive number is the sum of the previous number and a fixed difference. The fixed difference is called the common difference.
We have formed a linear equation in one variable in terms of \[x\] in the solution. A linear equation in one variable is an equation that can be written in the form \[ax + b = 0\], where \[a\] is not equal to 0, and \[a\] and \[b\] are real numbers. For example, \[x - 100 = 0\] and \[100P - 566 = 0\] are linear equations in one variable \[x\] and \[P\] respectively. A linear equation in one variable has only one solution. Therefore, there is only one value of \[x\] that satisfies the equation \[2x - 1 = x + 2\].