Question

if the Pth term of an AP is Q and its Qth term is P then show that its (P+Q)th term is zero/

Answer 1

The answer is given below :

Let us consider that the first term of the AP is a and the common ratio is d.

Given,

P-th term = Q

=> a + (P - 1)d = Q .....(i)

and

Q-th term = P

=> a + (Q - 1)d = P .....(ii)

We have

a + (P - 1)d = Q .....(i)
a + (Q - 1)d = P .....(ii)

On subtraction, we get

(P - 1 - Q + 1)d = Q - P

=> (P - Q)d = -(P - Q)

=> d = -1 [eliminating (P - Q)]

So, commn ratio (d) = -1

Putting d = -1 in (i), we get

a + (P - 1)(-1) = Q

=> a = P + Q - 1

So, first term = P + Q - 1

Therefore, the (P + Q)-th term is

= a + (P + Q - 1)d

= P + Q - 1 + (P + Q - 1)(-1)

= P + Q - 1 - P - Q + 1

= 0 [Proved]

Thank you for your question.

Answer 2

According to the Question

Let a be first term be a

And Common Difference be d

Therefore

$\bf\huge a_{p} = q , a_{q} = p$

a + (p - 1)d = q ……. (1)

a + (q - 1)d = p ……..(2)

Subtracting equations we get :-

(p - q)d = q - p

d = -1

Put the value of d in eq (1) :-

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

a = (p + q - 1)

$\bf\huge a_{p + q} = a + (p + q - 1)d$

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

= 0

Hence we get the (p + q)th term is Zero