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If $p^{\text{th}}$ term of an A.P. is $q$ and $q^{\text{th}}$ term is $p$, then prove that its $n^{\text{th}}$ term is $(p + q - n)$.
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$a_p = a + (p - 1)d = q$
quad (i)
$a_q = a + (q - 1)d = p$
quad (ii)
Solving (i) and (ii)
$d = -1, a = q + p - 1$
$a_n = (q + p - 1) + (n - 1)(-1) = q + p - n$
quad (i)
$a_q = a + (q - 1)d = p$
quad (ii)
Solving (i) and (ii)
$d = -1, a = q + p - 1$
$a_n = (q + p - 1) + (n - 1)(-1) = q + p - n$