Question

If the $t^{th}$ term of an AP is s and $s^{th}$ term of the same AP is t, then $a_n$ is

• A. t + s + n
• B. t + s - n
• C. t - s + n
• D. t - s - n

Answer 1

The correct option is B

t + s - n

Given,
$a_t$ = a + (t - 1)d = s . . . (i)
and $a_s$ = a + (s - 1)d = t . . . (ii)
Now,
(t - s) = a + (s - 1)d - a - (t - 1)d
$\Rightarrow$ t - s = (s - t)d
[from Eqs. (i) and (ii)]
$\Rightarrow$ (t - s) = -(t - s)d
$\Rightarrow$ d = - 1
From Eq. (i), we get
a + (t - 1) (-1) = s
$\Rightarrow$ a = s + t - 1
Now, $a_n$ = a + (n - 1) d
= s + t - 1 + (n - 1) $\times$ (-1)
= s + t - 1 - n + 1
$\therefore a_n$ = s + t - n