Answer 1

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 (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

⇒ a_m+n = 0