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

Last updated date: 18th Sep 2024
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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
$(5x + 2) - (4x - 1) = (4x - 1) - (x + 2) \\ \Rightarrow 5x + 2 - 4x + 1 = 4x - 1 - x - 2 \\ \Rightarrow x + 3 = 3x - 3 \\ \Rightarrow 2x = 6 \\ \Rightarrow x = 3 \\$
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.