Question

If $p^{th}$ term of an A.P. is q and is $q^{th}$ term is p. Show that its $r^{th}$ term is $(p+q-r)$

Answer 1

Given $p^{th}$ of an A.P is $q$ and $q^{th}$ term is $p$

We know that $n^{th}$ term of an A.P is $a+(n-1)d$

Where $a$ is the first term of the A.P and $d$ is the common difference of the A.P.

$⟹$$p=a+(q-1)d$ and $q=a+(p-1)d$

Substracting these two equations gives

$p-q=d(q-p)$

$⟹$$d=-1$

substituting $d=-1$ in one of the equations gives

$a=p+q-1$

$\therefore r^{th}termisp+q-1+(r-1)(-1)$

$⟹$ $p+q-r$

$\therefore r^{th}termisp+q-r$