nth partial sum derivation

don

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Mar 21, 2012, 5:11:18 PM3/21/12

to

In looking at the derivation of the nth partial sum equation for an
Arithmetic series which looks like this in my book:

Sn = a1 + (a1+d) + (a1+2d) + ..... (an -2d) + (an-d) + an
+ Sn = an + (an-d) + (an-2d) + .... (a1+2d + (a1+d) + a1
----------------------------------------------------------
2Sn = n ( a1+an )

Sn = n ( a1+an ) / 2

I understand the algebra but, I am just wondering why this trick of adding
something to itself in reverse order works? What was the logic behind this
formula and who was the first to realize this? I guess I am looking for the
rational behind this formula derivation. I hope my question makes sense. I
see something similar shown given for the formula behind finding the partial
sum for a geometric series..... which is also very clever.

William Elliot

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Mar 22, 2012, 3:52:07 AM3/22/12

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On Wed, 21 Mar 2012, don wrote:

> In looking at the derivation of the nth partial sum equation for an
> Arithmetic series which looks like this in my book:
>
> Sn = a1 + (a1+d) + (a1+2d) + ..... (an -2d) + (an-d) + an
> + Sn = an + (an-d) + (an-2d) + .... (a1+2d + (a1+d) + a1
> ----------------------------------------------------------
> 2Sn = n ( a1+an )
>
> Sn = n ( a1+an ) / 2
>

Huh? This isn't making sense. Let n = 3. So you mean
S3 = a1 + a2+d + a3+2d or S3 = a1 + a1+d +a1+2d?

Frederick Williams

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Mar 22, 2012, 11:08:58 AM3/22/12

to

don wrote:
>
> [...] What was the logic behind this

> formula and who was the first to realize this?

There is a well-known story about Gauss using it, but I suspect that it
is way, way older than that.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

Ken Pledger

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Mar 22, 2012, 4:14:36 PM3/22/12

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In article <jkdg65$n1o$1...@reader1.panix.com>, "don" <d...@panix.com>
wrote:

> In looking at the derivation of the nth partial sum equation for an
> Arithmetic series which looks like this in my book:
>
> Sn = a1 + (a1+d) + (a1+2d) + ..... (an -2d) + (an-d) + an
> + Sn = an + (an-d) + (an-2d) + .... (a1+2d + (a1+d) + a1
> ----------------------------------------------------------
> 2Sn = n ( a1+an )
>
> Sn = n ( a1+an ) / 2
>

> I understand the algebra but I am just wondering why this trick of adding

> something to itself in reverse order works? What was the logic behind this
> formula and who was the first to realize this? I guess I am looking for the

> rationale behind this formula derivation....

The most basic case a_n = n was known to the Greeks (but not in
modern notation, of course). Using pebble patterns, they found the
triangular number 1 + 2 + 3 + ... + n by combining two copies with
one reversed, then halving the resulting heteromecic number n(n + 1):
Wilbur Knorr, "The Evolution of the Euclidean Elements," 1975,
pp.150-151.

Ken Pledger.

JEmebius

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Mar 23, 2012, 6:02:27 PM3/23/12

to don

don wrote:
> In looking at the derivation of the nth partial sum equation for an
> Arithmetic series which looks like this in my book:
>
> Sn = a1 + (a1+d) + (a1+2d) + ..... (an -2d) + (an-d) + an

> + Sn = an + (an-d) + (an-2d) + .... (a1+2d) + (a1+d) + a1

> ----------------------------------------------------------
> 2Sn = n ( a1+an )
>
> Sn = n ( a1+an ) / 2
>
> I understand the algebra but, I am just wondering why this trick of adding
> something to itself in reverse order works? What was the logic behind this
> formula and who was the first to realize this? I guess I am looking for the
> rational behind this formula derivation. I hope my question makes sense. I
> see something similar shown given for the formula behind finding the partial
> sum for a geometric series..... which is also very clever.
>
>
>
>
>
>

Reply by Frederick Williams
---------------------------

[Incomplete quatation:]

>> [...] What was the logic behind this

>> formula and who was the first to realize this?
>

> There is a well-known story about Gauss using it, but I suspect that it
> is way, way older than that.
>

Continuation reply by Johan E. Mebius
-------------------------------------

Of course your questions make sense. They are metamathematical rather than mathematical questions;
please read on.

The legend goes that Gauss got sometime in his schoolboy's time the assignment "Add all integers
from 1 up to and including 100" and almost immediately gave the correct answer 5050.

His teacher would have asked him how he found the sum that fast, after which young Carl Friedrich
Gauss replied something like "Well, add the first and the last terms, making 101; the same for the
second and next-to-last, making 101 too. One can do this 50 times altogether; the result is 50 x 101
= 5050"

I did not verify this story; I will take it for granted for now. Why does this trick work? Just
because it works. There is no deep math logic behind it. Gauss was attentive and had a remarkably
strong intuition and sense for patterns and regularities, like all great scientists and mathematicians.

Ciao: Johan E. Mebius

don

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Mar 23, 2012, 6:43:03 PM3/23/12

to

> Why does this trick work? Just because it works. There is no deep math
> logic behind it. Gauss was attentive and had a remarkably strong intuition
> and sense for patterns and regularities, like all great scientists and
> mathematicians.
>
> Ciao: Johan E. Mebius

ok thanks... so it is just a trick... I was looking too deep :-)

Charles Brenner

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Mar 24, 2012, 4:33:05 PM3/24/12

to

On Mar 21, 2:11 pm, "don" <d...@panix.com> wrote:
> In looking at the derivation of the nth partial sum equation for an
> Arithmetic series which looks like this in my book:
>
> Sn = a1 + (a1+d) + (a1+2d) + ..... (an -2d) + (an-d) + an
> + Sn = an + (an-d) + (an-2d) + .... (a1+2d + (a1+d) + a1
> ----------------------------------------------------------
> 2Sn = n ( a1+an )
>
> Sn = n ( a1+an ) / 2
>
> I understand the algebra but, I am just wondering why this trick of adding
> something to itself in reverse order works? What was the logic behind this
> formula

For purposes of addition, it's an obvious idea to represent a set by
it's average. (That is, given a set of N elements whose average is A,
NA is the sum of the elements.) For a set whose elements are an
arithmetic progression, the average is the middle element (if any),
or, pursuing that minor insight, is the average of any two elements at
the same distance above and below the middle. Snaking the sequence so
as to put such pairs of elements into correspondence is thus just a
way to take advantage of that observation.

Mike Trainor

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Mar 25, 2012, 1:37:52 PM3/25/12

to

On Fri, 23 Mar 2012 23:02:27 +0100, JEmebius <jeme...@xs4all.nl>
wrote:

>don wrote:
>> In looking at the derivation of the nth partial sum equation for an
>> Arithmetic series which looks like this in my book:
>>
>> Sn = a1 + (a1+d) + (a1+2d) + ..... (an -2d) + (an-d) + an
>> + Sn = an + (an-d) + (an-2d) + .... (a1+2d) + (a1+d) + a1
>> ----------------------------------------------------------
>> 2Sn = n ( a1+an )
>>
>> Sn = n ( a1+an ) / 2
>>
>> I understand the algebra but, I am just wondering why this trick of adding
>> something to itself in reverse order works? What was the logic behind this
>> formula and who was the first to realize this? I guess I am looking for the
>> rational behind this formula derivation. I hope my question makes sense. I
>> see something similar shown given for the formula behind finding the partial
>> sum for a geometric series..... which is also very clever.

My fourth grade daughter used to call the technique and "pairng and
grouping". Her class was quite adept at handling sums of long sets
of numbers but collapsing those that added to multiples of 10, etc.
This was also a trick in their bag.

mt