rishi kesh
OP warned about not using the homework template
Please explain the above limit problem.i am able to understand last steps but can't get initial 4 steps.

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Homework Helper
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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.

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rishi kesh
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

scottdave
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?

Mentor
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. :)