(1+2+3+.......+(n-1))^2 + n^3 = (1+2+3+.......+n)^2 Is this correct?

Posted on 2011-02-11
Last Modified: 2012-06-22
I have seen somewhere that this is true,  can somebody help me in proving this?
Question by:mohet01
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
LVL 31

Accepted Solution

GwynforWeb earned 65 total points
ID: 34877235

(1+2+3+.......+(n-1))^2 + n^3

= (n(n-1)/2)^2 + n^3  

= (n^4 -2n^3 +n^2)/4 + n^3  

= (n^4 +2n^3 +n^2)/4


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

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

yes it is true

Assisted Solution

shadow77 earned 60 total points
ID: 34877256
Let k = 1+2+3 +...+(n-1).  Then, k = (n-1)*(1 + n-1)/2 [# of terms times average value].
And k = (n^2 - n)/2.

Left hand side (lhs) = k^2 +n^3.
lhs = ((n^2 - n)/2)^2 + n^3.
lhs = (n^4 - 2n^3 + n^2)/4 + n^3.
lhs = (n^4 - 2n^3 + n^2 + 4n^3)/4.
lhs = (n^4 + 2n^3 + n^2)/4.

Right hand side (rhs) = (k + n)^2
rhs = ((n^2 - n)/2 + n)^2.
rhs = ((n^2 - n + 2n)/2)^2.
rhs = ((n^2 + n)/2)^2.
rhs = (n^4 + 2n^3 + n^2)/4.

Thus, lhs = rhs.

Author Closing Comment

ID: 34878093

Featured Post

Independent Software Vendors: We Want Your Opinion

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Lithium-ion batteries area cornerstone of today's portable electronic devices, and even though they are relied upon heavily, their chemistry and origin are not of common knowledge. This article is about a device on which every smartphone, laptop, an…
When we purchase storage, we typically are advertised storage of 500GB, 1TB, 2TB and so on. However, when you actually install it into your computer, your 500GB HDD will actually show up as 465GB. Why? It has to do with the way people and computers…
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…
I've attached the XLSM Excel spreadsheet I used in the video and also text files containing the macros used below.…
Suggested Courses

615 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question