Question

If ${a_m}$ denotes the ${m^{th}}$ term of an A.P then ${a_m}$ is

1. $\dfrac{2}{{({a_{m + k}} + {a_{m - k}})}}$
2. $\dfrac{{({a_{m + k}} - {a_{m - k}})}}{2}$
3. $\dfrac{{({a_{m + k}} + {a_{m - k}})}}{2}$
4. None of these

Answer 1

Since the given problem is based on the concept of arithmetic progression. So we can use the formula of finding ${n^{th}}$term of a progression and further all the options are given in terms of ${a_{m + k}}$and ${a_{m - k}}$ so first we have to find ${a_{m + k}}$and ${a_{m - k}}$ terms using the formula for finding ${n^{th}}$term of a progression and rearranging the terms as per the given options.

Complete step-by-step solution:
Since we have to find ${m^{th}}$ term of an A.P that is we have to find ${a_m}$
Let a be the first term and d be the common difference of A.P
Then the nth term of A.P is given by
${a_m} = a + (m - 1)d - - - \left( 1 \right)$
Now let we can note that answer in the given options is in terms of ${a_{m + k}}$and ${a_{m - k}}$ so let us find ${a_{m + k}}$and ${a_{m - k}}$
${a_{m + k}}$can be obtained by replacing$m$
by $m + k$in equation (1) we get
${a_{m + k}} = a + (m + k - 1)d - - - (2)$
${a_{m - k}} = a + (m - k - 1)d - - - (3)$
Adding equation (2) and (3)
${a_{m + k}} + {a_{m - k}} = 2a + (m + k - 1 + m - k - 1)d$
On simplification we get
$\Rightarrow {a_{m + k}} + {a_{m - k}} = 2a + 2(m - 1)d$
Now taking 2 as common factor in the above equation we get
${a_{m + k}} + {a_{m - k}} = 2(a + (m - 1)d)$
Now RHS is of the form ${a_m}$
With this the above equation becomes
${a_{m + k}} + {a_{m - k}} = 2{a_m}$
Since we need the value of ${a_m}$so extract the value of ${a_m}$ we get
${a_m} = \dfrac{{{a_{m + k}} + {a_{m - k}}}}{2}$
Therefore, the correct answer is option 3) ${a_m} = \dfrac{{{a_{m + k}} + {a_{m - k}}}}{2}$.

Note: As arithmetic progression is a sequence where each term, except the first term, is obtained by adding a fixed number to its previous term, here, the “fixed number” is called the “common difference” and is denoted by d. The nth term of arithmetic progression depends on the first term and the common difference of the arithmetic progression.