Question

If 2, x, 26 are in arithmetic progression, find the value of $x$.

Answer 1

In this question, we are given that $2,x,26$ are in arithmetic progression and we have been asked to find the value of $x$. Find the common difference between the two pairs of numbers one by one and then equate them as the common difference is the same. Then, shift and find the value $x$.

Complete step-by-step solution:
We are given that $2,x,26$ are in arithmetic progression.
If a, b, c are in A.P, then $2b = a + c$. In this case, $a = 2,b = x,c = 26$.
Putting the values in the formula $2b = a + c$,
$\Rightarrow 2x = 2 + 26$
On adding and shifting we will get the value of x.
$\Rightarrow 2x = 28$
Shifting 2 to the other side,
$\Rightarrow x = \dfrac{{28}}{2} = 14$
Therefore, the value of $x$ is $14.$
We can also verify our answer. Since the given numbers form an arithmetic progression, the common difference will be equal. So, first we will find the common difference between the first two numbers and then between the last two numbers. If the difference is the same, the value of $x$ is correct.
Common difference between 2 and 14.
$\Rightarrow d = 14 - 2 = 12$
Now, we will find a common difference between 14 and 26.
$\Rightarrow d = 26 - 14 = 12$
Since the common difference is the same, the value of $x$ that we have found is correct.

Note: If you don’t remember the formula $2b = a + c$, then you can use the common difference to find the value of $x$. We have to use the same method to find the value of $x$ that we used above to verify our answer.
We know that $2,x,26$ are in arithmetic progression.
Finding the common difference,
$\Rightarrow {d_1} = x - 2$
$\Rightarrow {d_2} = 26 - x$
We know that their common differences must be the same. Hence, we will equate ${d_1}$ and ${d_2}$.
$\Rightarrow x - 2 = 26 - x$
Shifting to find the value of x,
$\Rightarrow x + x = 26 + 2$
This has brought us back to the equation we used above.
$\Rightarrow x = \dfrac{{28}}{2} = 14$
Hence, $x = 14.$