Question

If in A.P., p^th term is q and q^th term is p, then find (p + q)^th term.

Answer 1

Let first term of A.P. = a

and common difference = d

Then p^th term = T_p = a + (p – 1)d = q

⇒ a + (p – 1)d = q … (i)

and q^th term =  T_p = a + (q – 1)d = p

⇒ a + (q – 1)d = p   …. (ii)

(p + q^th term =) T_p+q

= a+ {(p + q) -1}  …. (iii)

Subtracting equation (ii) from (i),

{a + (p – 1)d} – {a + (q – 1)d} = q – p

⇒ a + (p – 1) d – a – (q – 1)d = q – p

⇒ (p – 1 – q + 1 )d = q – p

⇒ (p – q)d – (p – q)

⇒ d = -1

Putting value of d in equation (i)

a + (p- 1) (- 1) = q

⇒ a – (p – 1) = q

⇒ a = q + p – 1

⇒ a = p + q – 1

Putting the value of a and d in equation (iii),

^T_p+q = a + [(P + q) -]d

= (p + q – 1) + [(p + q -1)(- 1)]

= (p + q -1) – (p + q – 1)

= 0
Hence, (p + q)^th term is zero.