Question

If \[5,k,11\] are in AP, then the value of $k$ is:
A) $6$
B) $8$
C) $7$
D) $9$

Answer 1

Hint:
We are given three terms which are in Arithmetic Progression. We know that for arithmetic progression the common difference is the same for any two consecutive terms. We will use this fact and then find the value of the middle term that is asked.

Complete step by step solution:
It is given that the terms $5, k, 11$ are in an arithmetic progression.
We will consider that the arithmetic progression starts from the number $5$ itself.
We usually denote the first term of an arithmetic progression by $a$ so we will consider $a = 5$ in the given case.
Consider a general arithmetic progression with the first term $a$. The common difference between consecutive terms is constant so we will assume that the common difference is $d$.
Therefore, the next term in the assumed arithmetic progression will be $a + d$ .
Now again the common difference is the same so the third term in the same arithmetic progression is given by $\left( {a + d} \right) + d = a + 2d$ .
We will use a similar technique to determine the rest of the term in the arithmetic progression.
In general, the arithmetic progression is given by $a, a + d, a + 2d, a + 3d, \ldots ,a + \left( {n - 1} \right)d$.
Now we will consider the given arithmetic progression.
We have $a = 5$ and the common difference as $d$.
The second term in the arithmetic progression is given by $a + d = 5 + d$.
But it is given that the second term is $k$.
Therefore,
$5 + d = k$ … (1)
Similarly, the third term is given by $a + 2d = 5 + 2d$.
 But the third term given is $11$.
Therefore,
$5 + 2d = 11$
This implies that,
$2d = 6$
Simplifying for $d$ we get, $d = 3$.
Therefore, the common difference is $d = 3$.
Using this in equation (1) we get, $5 + 3 = k$.
Therefore, $k = 8$.

Hence, the correct option is B.

Note:
Here observe that the terms are in Arithmetic Progression. Therefore, common difference is the key factor here. We also considered that the Arithmetic progression given to us starts from the first given number itself as the properties of an arithmetic progression will not change in such a case. Finally, we just simplified the equation for finding the value of the unknown.