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State True/False.

If the pth term of an AP is q and its q th term is p then its $(p + q)$ th term is zero.

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True

False

answer is A.

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Detailed Solution

Given that

p t h

term of AP is q and

q t h

term of AP is p.
Let first term be a and common difference be d.
So, from first given condition: h

∴ a p = q

Formula for the

n t h

term:

a n = a + n − 1 × d

Applying the formula of nth term of A.P.

∴ a + (p − 1) d = q

-----(1)
Now, from second given condition:

∴ a q = p

Applying the formula of nth term of A.P.

∴ a + (q − 1) d = p

------(2)
Subtract equation (1) from equation (2):

∴ a + (q − 1) d − {a + (p − 1) d} = p − q ⇒ a + q d − d − {a + p d − d} = p − q ⇒ a + q d − d − a − p d + d = p − q ⇒ q d − p d = p − q

⇒ d (q − p) = p − q ⇒ d (q − p) = − (q − p) ⇒ d = − (q − p) q − p ⇒ d = − 1

Put the value of d in equation (1):

∴ a + (p − 1) × (− 1) = q ⇒ a − p + 1 = q ⇒ a = p + q − 1

Applying the formula of

n th

term of AP for

a p + q

∴ a p + q = (p + q − 1) + (p + q − 1) (− 1) ⇒ a p + q = (p + q − 1) − (p + q − 1) = 0

Therefore, the

(p + q) t h

term is 0.
Thus, the given statement is true.
Hence, option (1) is correct.

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