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If m times the m^{th} term of an Arithmetic Progression is equal to n times its n^{th} term and m ≠ n, show that the (m + n)^{th} term of the A.P. is zero.

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#### Solution

Let a be the first term and d is the common difference of an A.P

a_{m} and a_{n} be the m^{th} and n^{th }term respectively.

We have given m times the m^{th} term is equal to n times the n^{th} term

So, the equation becomes

m × a_{m} = n × a_{n}

We know that a_{m} = a + (m - 1)d

Similarly, a_{n }= a + (n - 1)d

m [a + (m - 1)d] = n [a + (n - 1)d]

m [a + (m - 1)d] - n [a + (n - 1)d] = 0

⇒ am + m (m - 1)d - an - n (n-1)d = 0

⇒ a (m - n) + [d (m^{2 }- n^{2}) - d (m - n )] = 0

⇒ a (m - n)+ d[(m - n) (m - n) - (m - n)] = 0

⇒ (m - n) [a + d ((m + n ) -1)] = 0

⇒ a + [(m + n) - 1]d = 0

⇒ a_{m+n} = 0

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