It is currently Mon May 21, 2018 6:06 pm

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Perfect squaresPosted: Fri Jan 15, 2016 10:32 pm
 Team Member

Joined: Mon Nov 09, 2015 1:36 am
Posts: 447
Location: Ioannina, Greece

_________________
Grigorios Kostakos

Top

 Post subject: Re: Perfect squaresPosted: Fri Jan 15, 2016 10:34 pm

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 831
Location: Larisa
Grigorios Kostakos wrote:

Hello Grigoris (Αρχιμήδης 6 greets you),

the numbers appearing in the sequence above are of the form:

\begin{aligned} a_k &=4\left ( 10^{2k-1}+10^{2k-2}+\cdots +10^k \right )+ 8\left ( 10^{k-1}+\cdots +10 \right )+9 \\ &=4\left ( 10^k \cdot \frac{10^k-1}{10-1} \right )+ 8 \left ( 10 \cdot \frac{10^{k-1}-1}{10-1}\right )+9 \\ &= \frac{4}{9}\left ( 10^{2k}-10^k \right )+\frac{8}{9} \left ( 10^k-10 \right )+9 \\ &= \left ( \frac{2\cdot 10^k +1}{3} \right )^2 \end{aligned}

that are indeed perfect squares. Also note that each term is an integer since $$\displaystyle 10^k \equiv 1 \bmod 3, \; \;\; 2 \cdot 10^k +1 \equiv 0 \bmod 3$$
Hence $\dfrac{2\cdot 10^k+1}{3} \in \mathbb{Z}$.

_________________
Imagination is much more important than knowledge.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 2 posts ]

 All times are UTC [ DST ]

#### Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta