$\displaystyle \begin{matrix} (1+x)(1+x)^k \ge (1+x)(1+kx) & \iff & (1+x)^{k+1} \ge 1+(k+1)x+kx^2 \end{matrix}.$
I don't understand the right side of inequation, shouldn't it be: $\displaystyle \ge 1+(k+1)x$
I see that the inequality must be strictly multiplied by "(1+x)" on both sides (in order to remain the same inequality), but shouldn't one, strictly following rules of mathematical induction, insert "k+1" on both sides, and thus get $\displaystyle (1+x)(1+x)^k \ge 1+(k+1)x$