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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.
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Solution
In the given AP, let the first be a and the common difference be d.
Then, Tn = a + (n – 1)d
⇒ Tp = a + (p – 1)d = q ...(i)
⇒ Tq = a + (q – 1)d = p ...(ii)
On subtracting (i) from (ii), we get:
(q – p)d = (p – q)
⇒ d = –1
Putting d = –1 in (i), we get:
a = (p + q – 1)
Thus, a = (p + q – 1) and d = –1
Now, Tp + q = a + (p + q – 1)d
= (p + q – 1) + (p + q – 1)(–1)
= (p + q – 1) – (p + q – 1) = 0
Hence, the (p + q)th term is 0 (zero).
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