A sum

Aug 2018
137
7
România
Hello all,

Calculate \(\displaystyle \sum_{k=n}^{k=1} k\).

All the best,

Integrator
 
Last edited:
Jun 2019
493
262
USA
That doesn't mean anything. Did you forget some terms?

The standard form is $\displaystyle \sum_{index~variable=starting~value}^{ending~value} (term~depending~on~index~variable)$
 
  • Like
Reactions: 1 person

skeeter

Math Team
Jul 2011
3,354
1,845
Texas
Hello all,

Calculate \(\displaystyle \sum_{k=n}^{k=1} k\).

All the best,

Integrator
so, assuming $n > 1$, is this what you mean?

$\displaystyle \sum_{k=n}^{k=1} k = n + (n-1) + (n-2) + \, ... \, + 3 + 2 + 1$

Doesn't this end up with the same result as \(\displaystyle \sum_{k=1}^n k\) ?


If this is not what you meant, what is the relationship between $n$ and $1$ ?
 
  • Like
Reactions: 1 person
Aug 2018
137
7
România
so, assuming $n > 1$, is this what you mean?

$\displaystyle \sum_{k=n}^{k=1} k = n + (n-1) + (n-2) + \, ... \, + 3 + 2 + 1$

Doesn't this end up with the same result as \(\displaystyle \sum_{k=1}^n k\) ?


If this is not what you meant, what is the relationship between $n$ and $1$ ?
Some say that \(\displaystyle \sum_{k=n}^{k=1} k\) it's not equal with \(\displaystyle \sum_{k=1}^{k=n} k\) where \(\displaystyle k,n\in \mathbb N\).Thank you very much!

All the best,

Integrator
 
Jun 2019
493
262
USA
Some say that \(\displaystyle \sum_{k=n}^{k=1} k\) it's not equal with \(\displaystyle \sum_{k=1}^{k=n} k\) where \(\displaystyle k,n\in \mathbb N\).
Well, who says that, and what do they say about it?

I've never seen reverse indexing like that in mathematics before. My first instinct would just be to count backwards, like skeeter.

My second instinct would be to interpret it like computer code for k = n to 1 step 1. If $k=1$, then sum the one term. if $k>1$, then no terms count and the sum is zero.
 
  • Like
Reactions: 1 person

skeeter

Math Team
Jul 2011
3,354
1,845
Texas
"Some say" that \(\displaystyle \sum_{k=n}^{k=1} k\) it's not equal with \(\displaystyle \sum_{k=1}^{k=n} k\) where \(\displaystyle k,n\in \mathbb N\).
Sounds like one of Trump's opening lines ... ;)
 
  • Like
Reactions: 1 person

topsquark

Math Team
May 2013
2,508
1,048
The Astral plane
Sounds like one of Trump's opening lines ... ;)
No, that's fake news. Fake news. I mean, it could be something, but I don't think it really is. Fake means not right, not right, but fake. Fake news. That's all it is. Fake. How does my tie look? I tied it myself. It looks great. Really, I think it does. I like it.

-Dan
 
  • Like
Reactions: 1 person
Aug 2018
137
7
România
Well, who says that, and what do they say about it?

I've never seen reverse indexing like that in mathematics before. My first instinct would just be to count backwards, like skeeter.

My second instinct would be to interpret it like computer code for k = n to 1 step 1. If $k=1$, then sum the one term. if $k>1$, then no terms count and the sum is zero.
Hello,

It is true that the calculation of a sum is somewhat like the calculation of an integral? I say yes and then we can write that \(\displaystyle \sum_{k=1}^{k=n} k=\frac{n+1}{2}+\int_1^nx dx\), which means that \(\displaystyle \sum_{k=n}^{k=1} k=\frac{n+1}{2}+\int_n^1x dx=\frac{n+1}{2}+\frac{1-n^2}{2}=\frac{-n^2+n+2}{2}=-\frac{(n-2)(n+1)}{2}\). Is my reasoning correct?

All the best,

Integrator
 
Last edited by a moderator:

skipjack

Forum Staff
Dec 2006
21,468
2,464
which means that
Why?

I can't see how you could justify the term \(\displaystyle \frac{n + 1}{2}\) in your first equation in such a way that it would still be justified in the second equation.
 
  • Like
Reactions: 1 person