Hi **bmp05**.

Originally Posted by

**bmp05** ii) but then... ?

$\displaystyle c \left(\sum_{k=1} ^n x_n \right) = \sum_{k=1} ^n c x_n = c x_1 + c x_2 + c x_3 + ... + c x_n $

I mean... what am I showing here, exactly, $\displaystyle P(n-1) \Rightarrow P(n)$?

$\displaystyle c \left(\sum_{k=1} ^n x_n \right) = \sum_{k=1} ^{n-1} c x_{n-1} + c x_n$

For the inductive step, you assume that

$\displaystyle c \left(\sum_{k=1} ^n x_n \right) = \sum_{k=1} ^n c x_n\quad\ldots\,\fbox1$

is true, then you try and prove that $\displaystyle c \left(\sum_{k=1} ^{n+1} x_n \right) = \sum_{k=1} ^{n+1} c x_n$ is also true. Can you do that?

Hint: Add $\displaystyle cx_{n+1}$ to both sides of $\displaystyle \fbox1.$