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If $m^{th}$ term of an A.P. is $n$ and $n^{th}$ term is $m$ , find the $r^{th}$ term.

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m + n - r

m + n + r

m - n + r

r - (m + n)

answer is A.

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Detailed Solution

Given,

m^{th}

term of an A.P. is

n

and

n^{th}

term is

m

.
The

n^{th}

term of an A.P is calculated using

a_{n} = a + (n - 1) d

.
The

r^{th}

the term of an A.P is
a = a, n = r,

a_{r} = a + (r - 1) d

.
Substitute values for mth term,

a_{m} = a + (m - 1) d

a + (m - 1) d = n

….(i)
Substitute values for nth term,

a + (n - 1) d = m

….(ii)
Solve equation (i) and (ii),

\Rightarrow (n - m) = (m - 1) d - (n - 1) d

\Rightarrow (n - m) = md - d - nd + d

\Rightarrow (n - m) = (md - nd)

\Rightarrow (n - m) = - d (n - m)

\Rightarrow d = - 1

Substitute the value in equation (i),

\Rightarrow a + (m - 1) (- 1) = n

\Rightarrow a - m + 1 = n

\Rightarrow a = m + n - 1

Then

(r)^{th}

term is,

\Rightarrow a_{r} = a + (r - 1) d

\Rightarrow a_{r} = n + m - 1 + (r - 1) (- 1)

\Rightarrow a_{r} = n + m - 1 - r + 1

\Rightarrow a_{r} = n + m - r

Hence rth term is

n + m - r

Therefore, option 1 is correct.

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