Please explain this limit problem

[rishi kesh]

Please explain the above limit problem.i am able to understand last steps but can't get initial 4 steps.

 

[scottdave]

The sum of natural numbers 1 to N. Is that the part you are confused about?
Let's do an example. Say you want to add the numbers 1 to 10. You can group them: Take the first number and the last number (1 + 10) then 2nd number and next to last (2 + 9), then (3 + 8) etc. Each grouping adds to 11 {N+1}. There are 5 groupings {N/2}. So in general, we have (N + 1) * (N/2). This works with odd N values as well.

 

[rishi kesh]

The sum of natural numbers 1 to N. Is that the part you are confused about?
Let's do an example. Say you want to add the numbers 1 to 10. You can group them: Take the first number and the last number (1 + 10) then 2nd number and next to last (2 + 9), then (3 + 8) etc. Each grouping adds to 11. There are 5 groupings. So in general, we have (N + 1) * (N/2). This works with odd N values as well.

This trick is something interesting to know about. I will remember it.appreciate your help

 

[rishi kesh]

This trick is something interesting to know about. I will remember it.appreciate your help

But of you look at the 4th step they replaced 'n' by n-1 ..how do that work?

 

[DrClaude]

But of you look at the 4th step they replaced 'n' by n-1 ..how do that work?

The comment appears to be misplaced. The replacement occurs one step above, going from $\sum_{r=1}^n$ to $\sum_{r=1}^{n-1}$.

 

[rishi kesh]

The comment appears to be misplaced. The replacement occurs one step above, going from $\sum_{r=1}^n$ to $\sum_{r=1}^{n-1}$.

Oh i got it. :)