Factorial and induction

Question

Part of step in induction:

$(2 k+1)*((1*3* 5*\dotsb* (2k-1)) =1*3*5*\dotsb*(2k+1)$

Am I correct with believing that we in first instance went up to $k$, and then we went further to $2((k+1)-1$ hence the last expression?

Answer 1

Another way to write that is

\begin{align*} \left[\prod_{i=1}^k (2i-1)\right]\times(2k+1) &= \prod_{i=1}^{k+1} (2i-1) \\ &= 1\cdot3\cdot5\dotsb(2(k-1)-1)\cdot(2(k)-1)\cdot(2(k+1)-1)\\ &= 1\cdot3\cdot5\dotsb(2k-2-1)\cdot(2k-1)\cdot(2k+2-1)\\ &= 1\cdot3\cdot5\dotsb(2k-3)\cdot(2k-1)\cdot(2k+1) \end{align*}

This way you can see/say that it "goes to $k+1$". So, yes, you are correct in believing that.