Question of RS Aggarwal book based on Arithmetic Progression For ICSE & CBSE Board Students.

Here the two *n*th term of AP is given you have to show that its both *n*th term Addition is Zero.

Question Number 26 Of Exercise 11A of RS Aggarwal Solution.

# If the pth term of an AP is q and its qth term is p then show that is (p+q)th term is zero.

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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,

common 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]