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Question

if m times the mth term is equal to n times the nth term. then what is (m+n)th term of an a.p

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Solution

Dear Student!

If the question is “if n times the mth term is equal to m times the nth term, then what is (m + n)th term of the A.P.”, then the answer is given below.
Let the first term and common difference of the A.P. be ‘a’ and ‘d’ respectively.
Then, mth term = a + (m – 1) d and nth term = a + (n – 1) d
By the given condition,
n [a + (m – 1) d] = m [a + (n – 1) d]
na + n (m – 1) d = ma + m (n – 1) d
n(m – 1)d m (n – 1)d = mana
d[n(m – 1) – m (n – 1)] = a (m – n)
d[nmnmn + m] = a (m – n)
d(m – n) = a (m – n)
d = a
Now (m + n)th term = a + (m + n – 1)d.
But d = a
∴ (m + n)th term = a + (m + n – 1) a = a + a(m + n) – a = a (m + n)

Best wishes!


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