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2016-04-25T11:53:05+05:30

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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.

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2016-04-25T14:52:45+05:30
We know that nth term  t_{n} = a+(n-1)d

m times  t_{m} =  n times  t_{n}
 
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

 t_{m+n} = 0
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