If the $t^{th}$ term of an AP is $s$ and $s^{th}$ term of the same AP is $t,$ then $a_{n}$ is ___.

t + s + n

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t + s - n

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t - s + n

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t - s - n

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Solution

The correct option is B $t + s - n$
Given,
$a_{t} = a + (t - 1) d = s, . . . (i)$
$a_{s} = a + (s - 1) d = t . . . (i i)$
$\Rightarrow (t - s) = a + (s - 1) d - a - (t - 1) d$
$\Rightarrow t - s = (s - t) d [from eqns. (i) and (ii)]$
$\Rightarrow (t - s) = - (t - s) d$
$\Rightarrow d = - 1$
From $(i),$ we get
$a + (t - 1) (- 1) = s .$
$\Rightarrow a = s + t - 1$
$a_{n} = a + (n - 1) d$
$\Rightarrow a_{n} = s + t - 1 + (n - 1) \times (- 1) = s + t - 1 - n + 1 = s + t - n$

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