Search   Memberlist   Usergroups
 Page 1 of 1 [2 Posts]
Author Message
Gottfried Helms
science forum Guru

Joined: 24 Mar 2005
Posts: 301

Posted: Thu Jul 13, 2006 2:40 am    Post subject: Re: Odd Squares, 8 (2^3), and the Sum of the First n Positive Integers

Am 13.07.2006 03:58 schrieb rer:
 Quote: I did see it noted that the odd squares are 1 (mod (http://mathworld.wolfram.com/SquareNumber.html). I imagine it is noted elsewhere, but they also have a relationship to the sum of the first n positive integers. If it is noted that the sum of the first n positive integers = n(n+1)/2, then ((2^3)(((n(n+1))/2))+1=(2n+1)^2 (8(n^2+n)/2)+1=4(n^2)+2n+2n+1 (4(n^2+n))+1=4(n^2)+4n+1 4(n^2)+4n+1=4(n^2)+4n+1

.......... = (2n +1)^2 by binomial expansion.

Gottfried Helms
rer
science forum beginner

Joined: 10 Jul 2005
Posts: 21

Posted: Thu Jul 13, 2006 1:58 am    Post subject: Odd Squares, 8 (2^3), and the Sum of the First n Positive Integers

I did see it noted that the odd squares are 1 (mod
(http://mathworld.wolfram.com/SquareNumber.html). I imagine it is noted
elsewhere, but they also have a relationship to the sum of the first n
positive integers.

If it is noted that the sum of the first n positive integers =
n(n+1)/2, then

((2^3)(((n(n+1))/2))+1=(2n+1)^2

(8(n^2+n)/2)+1=4(n^2)+2n+2n+1
(4(n^2+n))+1=4(n^2)+4n+1
4(n^2)+4n+1=4(n^2)+4n+1

for example:

the sum of the first positive integer, 1, is 1
then, (8*1)+1=9 or ((2*1)+1)^2=3^2

the sum of the first two positive integers, 1+2, is 3
then, (8*3)+1=25 or ((2*2)+1)^2=5^2

the sum of the first three positive integers, 1+2+3, is 6
then, (8*6)+1=49 or ((2*3)+1)^2=7^2

the sum of the first four positive integers, 1+2+3+4, is 10
then, (8*10)+1=81 or ((2*4)+1)^2=9^2

the sum of the first five positive integers, 1+2+3+4+5, is 15
then, (8*15)+1=121 or ((2*5)+1)^2=11^2

etc.

 Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
 Page 1 of 1 [2 Posts]
 The time now is Thu Feb 21, 2019 8:02 am | All times are GMT
 Jump to: Select a forum-------------------Forum index|___Science and Technology    |___Math    |   |___Research    |   |___num-analysis    |   |___Symbolic    |   |___Combinatorics    |   |___Probability    |   |   |___Prediction    |   |       |   |___Undergraduate    |   |___Recreational    |       |___Physics    |   |___Research    |   |___New Theories    |   |___Acoustics    |   |___Electromagnetics    |   |___Strings    |   |___Particle    |   |___Fusion    |   |___Relativity    |       |___Chem    |   |___Analytical    |   |___Electrochem    |   |   |___Battery    |   |       |   |___Coatings    |       |___Engineering        |___Control        |___Mechanics        |___Chemical

 Topic Author Forum Replies Last Post Similar Topics Distribution of Goldbach pairs as even integers n increases stargene@sbcglobal.net Math 1 Mon Jul 17, 2006 5:02 am Squares Gerry Math 11 Thu Jul 13, 2006 7:50 pm Percentages from/to negative integers WhytheQ Math 3 Fri Jul 07, 2006 8:55 am Questions on Least Squares Fitting for Logarithmic Relations draccarlawpet Math 9 Thu Jul 06, 2006 6:08 pm random selections of integers from the infinite set of al... Pete B Math 42 Thu Jul 06, 2006 3:59 pm