Sir please give me a detailed solution of this question in which we have been asked to show that its (p + q)th term is zero, if the pth term of an AP is q and its q term is p.
RS Aggarwal, class 10, chapter 5A, question no 40.
Rajan@2021Guru
If the pth term of an AP is q and its q term is p then show that its (p + q)th term is zero.
Share
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]