I am currently studying proving by induction but I am faced with a problem.
I need to solve by induction the following question.
$$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$
for all $n > 1$.
Any help on how to solve this would be appreciated.
This is what I have done so far.
Show truth for $N = 1$
Left Hand Side = 1
Right Hand Side = $\frac{1}{2} (1) (1+1) = 1$
Suppose truth for $N = k$
$$1 + 2 + 3 + ... + k = \frac{1}{2} k(k+1)$$
Proof that the equation is true for $N = k + 1$
$$1 + 2 + 3 + ... + k + (k + 1)$$
Which is Equal To
$$\frac{1}{2} k (k + 1) + (k + 1)$$
This is where I'm stuck, I don't know what else to do. The answer should be:
$$\frac{1}{2} (k+1) (k+1+1)$$
Which is equal to:
$$\frac{1}{2} (k+1) (k+2)$$
Right?
By the way sorry about the formatting, I'm still new.
Answer
Basic algebra is what's causing the problems: you reached the point
$$\frac{1}{2}K\color{red}{(K+1)}+\color{red}{(K+1)}\;\;\;\:(**)$$
Now just factor out the red terms:
$$(**)\;\;\;=\color{red}{(K+1)}\left(\frac{1}{2}K+1\right)=\color{red}{(K+1)}\left(\frac{K+2}{2}\right)=\frac{1}{2}(K+1)(K+2)$$
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