Finding my way

The sum of the consecutive numbers starting from 1 to any number N is N*(N+1)/2. I was reading the proof of this in some blog or article. It went like this:

1 + 2 + 3 + .... N
Rearranging so that the higher numbers are alongside the lower numbers, we get
1 + N + 2 + N-1 + 3 + N-2 + ...
Now since the number of terms will differ if N is either odd or even, we consider each case separately.
If N is even:
(1+N) + (2+N-1) + (3+N-2) + .... (N/2 + N/2 + 1) This has now N/2 groupings
The above expression can be viewed as N/2 occurrences of the the term (N+1)
So the value of he expression = (N/2) * (N+1) = N*(N+1)/2
If N is odd:
then we group together all terms except the middle term which is (N+1)/2 .
(1+N) + (2+N-1) + (3+N-2) + ....+ (N+1)/2
The above expression can be viewed as (N-1)/2 groupings of the term (N+1) plus the middle term (N+1)/2
The value of the expression = (N+1) * (N-1)/2 + (N+1)/2
= (N+1)/2 [N - 1 +1] = N*(N+1)/2

So we have proved that the sum of consecutive numbers from 1 to N is N*(N+1)/2, regardless of whether N is even or odd.

This led me to think - the result is regardless of whether N is even or odd, so shouldn't there way to calculate this expression without bringing 2 separate cases (when N is even and when is odd) into the picture?

So lets see if we can calculate the value, without applying any strategies that will force us to consider the two separate cases.
Let S denote the sum of the consecutive numbers from 1 to N.
S = 1 + 2 + 3 + .... N
Lets write the term on the right hand side backwards, counting from the highest term to the lowest.
S = N + N-1 + N-2 + N-3 + .... + N-(N-2) + N-(N-1)
Group all the Ns together
S = N + N + N +N .... -1 -2 -3 ....- N-2 - N-1
S = N + N + N +N .... - (1 + 2 + 3 ....+ N-2 + N-1)
Look at the grouped term furthest to the right. Its actually the sum of N consecutive terms minus the last term N! So we can write it as:
S = N*N - (S - N)
S = N*N - S + N
2S = N*N + N
2S = N*(N+1)
S = N*(N+1)/2

So there we have it. We have proved that the sum of consecutive numbers from 1 to N is N*(N+1)/2 whether N is even or odd.

Now, lets get to what I REALLY wanted to say. The fact is, when I was in school, I would have never applied the reasoning like I applied it here. I wouldn't go searching for a proof just because I knew it should exist. Strangely, it seems that if I tackle my math and physics problems in school now, I feel I would be more successful at it. Thats because I feel my reasoning is much more mature, plus the fact that I view it as an enjoyable, leisurely activity. I am somewhat puzzled at how people think that as they grow older, they do not have the mental capabilities of solving problems that they had when they were in school. To me, I believe that you can actually increase your capacity to solve problems once you mature. This strengthens my belief in the theory that if we keep exercising our brain, our capacity to think clearly increases. Its not a question of age. Age might play a part much much later, (if ones about 80 or so maybe, when your brain retires, like everybody else ;)) but if anybody says they cant solve problems as good as they did before, they are just not making it a point to exercise their brain enough.

So go and find ways to test and exercise your brain. If you think you are in a job that doesnt give you enough opportunities to do that, make time to solve some puzzles, or draw something or take some music or art classes - anything that makes you think.