(15) If m times the mth term of an A.P is equal to the n times the nth term, then show that the (m + n)th term of the A.P is zero

Solution:
m tm = n tn
m [ a+ (m - 1) d ] = n [ a+ (n - 1) d ]
m a + m d (m – 1)  = n a + n d (n – 1)
m a + m² d – m d  = n a + n² d – n d
m a - n a + m² d - n² d – m d  + n d = 0
a (m – n) + (m² - n² – m + n ) d = 0
a (m – n) + [ (m + n) (m - n) –( m - n ) ] d = 0
a (m – n) + (m – n) [ (m + n) – 1 ] d = 0
Divide by (m - n) => a + [ (m + n) – 1 ] d = 0
                                 t (m + n) = 0

Therefore the (m+n) th term of the A.P is zero.