# If m times the mth term of an A.P. is n times its nth term,show that (m+n)th term of the A.P. is zero

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Tm= n and Tn =m
d= Tp-Tq/ p-q =
Tm-Tn/ m-n = n-m / m-n = take - common ( m-n) / m-n = -1 consider
Tm= n
a+(m-1) d=n
a-m+1= n
a+1= m+n ( equation 1)
Tn=n ( solve it same
by above method) Tm+n = a+ ( m+n-1) d =
a-m-n+1= a+1= m-n = m+n=m-n=0 hence we
proved.

m times = n times

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

ma + m²d - md = na + n²d - nd

ma - na + m²d - n²d +md - nd = 0

a(m-n) + d(m²-n²) + d(m-n) = 0

taking (m-n) common and simplifying

a + d(m+n) - d = 0

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