Help with induction step

[sk1234]

Hello everyone,

I was following an induction example and - as is often the case for me, unfortunately - I got stuck in one particular step where I don't understand how the author got from A to B.

Here it is:
[math]\frac{(-1)^k k(k+1)}{2}+(-1)^{k+1} (k+1)^2[/math][math]=\frac{(-1)^k (k+1)}{2}(k-2(k+1))[/math]
I don't understand how the equation got from one step to the next.
Could someone please break it down for me? If you have tips about where to learn more about these intermediate steps in induction proofs, I would also really appreciate it.
Thanks in advance.

 

[Deleted member 4993]

Hello everyone,

I was following an induction example and - as is often the case for me, unfortunately - I got stuck in one particular step where I don't understand how the author got from A to B.

Here it is:
[math]\frac{(-1)^k k(k+1)}{2}+(-1)^{k+1} (k+1)^2[/math][math]=\frac{(-1)^k (k+1)}{2}(k-2(k+1))[/math]
I don't understand how the equation got from one step to the next.
Could someone please break it down for me? If you have tips about where to learn more about these intermediate steps in induction proofs, I would also really appreciate it.
Thanks in advance.

[math]\frac{(-1)^k k(k+1)}{2} + (-1)^{k+1} (k+1)^2[/math]
We know:

[math](-1)^{k+1} (k+1)^2 \ \ = \ (-1) \ * \ (-1)^{k} (k+1)^2 \ \ [/math]
Then:

[math]\frac{(-1)^k k(k+1)}{2} + (-1)^{k+1} (k+1)^2[/math]
= [math]\frac{(-1)^k k(k+1)}{2} - (-1)^{k} (k+1)^2[/math]
= [math](-1)^k (k+1)\frac{k - 2(k+1)}{2} [/math]
continue.....

 

[Steven G]

If you have tips about where to learn more about these intermediate steps in induction proofs, I would also really appreciate it.
Thanks in advance.

You just have to know your algebra and make things work.