# 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.

Let a be the first term and d be common difference.

Given: ap = q

aq = p

To show: a(p + q) = 0

We know, nth term of an AP is
an = a + (n - 1)d
where, a is first term and d is common difference
Consider ap = q

a + (p - 1)d = q     (1)

Consider aq = p

a + (q - 1)d = p      (2)

Now, subtracting equation (2) from equation (1), we get

(p - q)d = (q - p)

d = - 1

From equation (1), we get,

a - p + 1 = q

p + q = a + 1 ……………………….(3)

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

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

= a + (a + 1 - 1)(-1)

(putting the value of p + q from equation 3)

= a + (-a)

= 0

a(p + q) = 0

Hence, proved.

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