Stuck in my induction proof

Question

I am at a step in my induction proof where I need to show that:

$\frac14(k+1)^{2}k^{2}+(k+1)^{3}$

is equal to

$\frac14(k+2)^{2}(k+1)^{2}$

Which I can't really seem to figure out how to. Can you help me out?

score 0 · Answer 1 · answered Oct 05 '18 at 18:31

0

Write $$(k+1)^2(\frac{1}{4}k^2+k+1)=\frac{1}{4}(k+1)^2(k^2+4k+4)=…$$

answered Oct 05 '18 at 18:31

mrtaurho · Accepted Answer · 2018-10-05T18:48:00.423

0

$$\begin{align} \frac14(k+1)^2k^2+(k+1)^3=\frac14((k+1)^2k^2+4(k+1)^3)&=\frac14((k+1)^2(k^2+4k+4))\\ &=\frac14(k+1)^2(k+2)^2 \end{align}$$

edited Oct 05 '18 at 18:48

answered Oct 05 '18 at 18:31

mrtaurho

Hm, I can't seem to understand where the factor 4 comes from in the second step. – Boris Grunwald Oct 05 '18 at 18:43
Ops. Sorry I missed the plus sign within the first equation. The $4$ appear while adding up the two terms $\frac14(k+1)^2k^2$ and $\frac44(k+1)^3$ which together yields to $\frac14((k+1)^2k^2+4(k+1)^3)$ – mrtaurho Oct 05 '18 at 18:49
Thanks, I think it makes sense now. – Boris Grunwald Oct 05 '18 at 18:52

score 0 · Answer 3 · answered Oct 05 '18 at 18:47

0

$$\frac14(k+1)^{2}k^{2}+(k+1)^{3}= (k+1)^2 \big ((1/4) k^2 +k+1) $$

$$= (k+1)^2 \big ( (1/4) (k^2+4k +4)\big ) =(1/4)(k+1)^2 (k+2)^2$$

answered Oct 05 '18 at 18:47

3 Answers3