Question

Find the value of $x$ for which $(5x+2)$,$(4x-1)$, and $(x+2)$are in A. P.

Answer 1

Hint: Using the definition of A.P. consider the difference between successive terms.Solve the equation $(5x+2)-(4x-1)=(4x-1)-(x+2)$ to obtain the value of $x$.


Complete step by step solution:
We are given three terms $(5x+2)$,$(4x-1)$, and $(x+2)$
These terms are in A. P., i.e., they are in Arithmetic Progression.
We are asked to find the value of the variable $x$.
We say that a given sequence of n numbers or terms $x_1,x_2,...x_n$ are in arithmetic progression when the difference between any two successive terms, called the common difference, is a constant.
Let d be the common difference.
Then, we have $$d=x_2-x_1=x_3-x_2=...=x_n-x_{n-1}$$
Using this definition, we can conclude that the difference between the successive terms of the sequence $(5x+2)$,$(4x-1)$, and$(x+2)$is a constant.
Therefore, we get
$$\begin{gathered} (5x+2)-(4x-1)=(4x-1)-(x+2) \\ \Rightarrow 5x+2-4x+1=4x-1-x-2 \\ \Rightarrow x+3=3x-3 \\ \Rightarrow 2x=6 \\ \Rightarrow x=3 \end{gathered}$$
Hence the value of $x$ is 3.


Note: The difference must be considered for successive terms. That is the order of the terms matter while solving problems related to A.P.