Question

a+(a+d) +(a+2d) + ...... +(a+(n-1)d)= 1/2 n× {2a+(n-1)d}, n belongs to N

Answer 1

Proof by PMI
Assume p(k) is true
a+a+d+(a+2d)………..[a+(k-1)d]+ [a+kd ] =k/2[2a+(k-1)d]+ [a+kd ]

=k(2a+kd-d)+2(a+kd)/2

=2ak+kd-kd+2a+2kd/2

=2ak+2a+kd+kd/2

=2a(k+1)+kd(k+1)/2

=(k+1)(2a+kd)/2

=(k+1)/2(2a+kd)
It is same as our assumption where k is replaced by k+1. Therefore it is true for all natural numbers