question_answer 141)

Use P.M.I. to prove that  

Answer:

(i) For n = 1              given hypothesis is true for n = 1. (ii) Let given hypothesis is true for n = k                        ? (1) (iii)  For n = k + 1             L.H.S.                         (Product rule)       = xk × 1 = x × xk             = xk + x  × k.xk?1      (by (1))       = xk + k.xk = xk (k + 1)        = R.H.S.       given hypothesis is true for n = k + 1 hence by P.M.I., Given hypothesis is true  Hence the result.