Question

Find the value of $x$ for which $\left( {8x + 4} \right),\left( {6x - 2} \right){\rm{ and }}\left( {2x + 7} \right)$ are in A.P.

Answer 1

Hint: When three numbers $a$, $b$ and $c$ are in arithmetic progression, they have a common difference between them i.e. $b - a = c - b$. The relation between the $a$, $b$ and $c$ when they are in A.P. is given by $b = \dfrac{{a + c}}{2}$. In A.P when the sum of the first and the last term is divided by 2 and multiplied by the total number of terms, the result gives the sum of all the terms in that A.P. series.

Complete step-by-step answer:
We know that when 3 numbers $a,b,c$ are in A.P. the common difference between them is equal.
This means, $b - a = c - b$.
So,
$
2b = a + c\\
b = \dfrac{{a + c}}{2}
$
As per the question, it is given that the three numbers $\left( {8x + 4} \right),\left( {6x - 2} \right){\rm{ and }}\left( {2x + 7} \right)$ are in A.P.
Now, we can compare
$\Rightarrow \left( {8x + 4} \right),\left( {6x - 2} \right){\rm{ and }}\left( {2x + 7} \right)$ with $a,b,c$.
On comparing, we get, $a = 8x + 4$, $b = 6x - 2$ and $c = 2x + 7$.
Now, we can substitute the value of $a = 8x + 4$, $b = 6x - 2$ and $c = 2x + 7$ in the equation $b = \dfrac{{a + c}}{2}$ to find the value of variable x.
On substituting the values, we get,
$\Rightarrow \left( {6x - 2} \right) = \dfrac{{\left( {8x + 4} \right) + \left( {2x + 7} \right)}}{2}$
Now, we should multiply both sides by 2.
 $2 \cdot \left( {6x - 2} \right) = \left( {8x + 4} \right) + \left( {2x + 7} \right)$
Multiply $6x - 2$ by 2 on the left hand side and add all terms on the right-hand side.
$\Rightarrow 12x - 4 = 10x + 11$
After subtracting by $10x$ on both sides of the equation, we get,
$
\Rightarrow 12x - 10x - 4 = 10x - 10x + 11\\
\Rightarrow 2x - 4 = 11
$
Now, we have to add 4 on both sides of the equation.
On adding 4 on both sides of equation, we get,
$
\Rightarrow 2x - 4 + 4 = 11 + 4\\
\Rightarrow 2x = 15
$
On dividing the entire equation by 2, we get,
$
\Rightarrow \dfrac{{2x}}{2} = \dfrac{{15}}{2}\\
\Rightarrow x = 7.5
$

Therefore, the value of $x$ is 7.5

Note: Students often make mistakes while using the formula for A.P. Instead of remembering it, students should understand the concept behind arithmetic progression. Consider $a$ as a term in A.P. series and $d$as the common difference between the terms of A.P. series, then we can say that $a - d$, $a$, $a + d$ are the three terms of an A.P series.