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If m times the mth term of an Arithmetic Progression is equal to n times its nth term and m ≠ n, show that the (m + n)th term of the A.P. is zero. - Mathematics

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Sum

If m times the mth term of an Arithmetic Progression is equal to n times its nth 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
am and an be the mth and nth term respectively.
We have given m times the mth term is equal to n times the nth term
So, the equation becomes

m × am = n × an

We know that am = a + (m - 1)d

Similarly, an = 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 (m2 - n2) - 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

⇒ am+n = 0

Concept: nth Term of an AP
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