Question

If the $p^{th}$ term of an A.P. is q and the $q^{th}$ term is p, prove that its $n^{th}$ term is (p + q – n).

Answer 1

Let a be the first term and d be the common difference of the given A.P. Then,

$p^{th}$ term = q

$a+(p–1)d=q$ .... (i)

$q^{th}$ term = p

$a+(q–1)d=p$ .... (ii)

Subtracting equation (ii) from equation (i), we get

$(p–q)d=(q–p)$

$d=–1$

Putting d = – 1 in equation (i), we get

$a=(p+q–1)$

$n^{th}$ term $=a+(n–1)d$

$=(p+q–1)+(n–1)\times (–1)$

$=(p+q–n)$