Question

tn=m and tm =n .show that tr=m+n-r

Answer 1

Answer:

t(m+n)= 0

Step-by-step explanation:

tm= n

∴n= a+(m-1)d....(1)

tn= m

∴m= a+(n-1)d....(2)

(1)-(2)⇒

∴n-m= a-a+md-d-nd+d

∴n-m= (m-n)d

∴d= (n-m)/(m-n)

Multiply the numerator and the denominator by (-1)

∴d= (n-m)(-1)/(m-n)(-1)

∴d= (n-m)(-1)/(n-m)

∴d= -1....(3)

Putting (3) in (1)

∴n= a+(m-1)(-1)

∴n= a+1-m

∴m+n= a+1

∴t(m+n)= t(a+1)

∴t(a+1)= a+(a+1-1)d

∴t(a+1)= a+a(-1)

∴t(a+1)= a-a

∴t(a+1)= 0

∴t(m+n)= 0 (∵t(m+n)= t(a+1))