[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include(/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php) [function.include]: failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include() [function.include]: Failed opening '/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php' for inclusion (include_path='.:/opt/php53/lib/php')
[phpBB Debug] PHP Warning: in file [ROOT]/includes/session.php on line 1042: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4786: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4788: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4789: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4790: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
BdMO Online Forum • View topic - Easy FE

Easy FE

For discussing Olympiad Level Algebra (and Inequality) problems
Facebook Twitter

Easy FE

Post Number:#1  Unread postby Absur Khan Siam » Tue Feb 28, 2017 12:22 pm

Find all function $f: N \rightarrow N$ such that ,
$f^3(1) + f^3(2) + ... + f^3(n) = (\frac{f(n)f(n+1)}{2})^2$ and $f(1) = 1$
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid
Absur Khan Siam
 
Posts: 53
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka

Re: Easy FE

Post Number:#2  Unread postby Absur Khan Siam » Tue Feb 28, 2017 12:37 pm

$f^3(1) + f^3(2) + ... + f^3(n) = (\frac{f(n)f(n+1)}{2})^2...(i)$
$f^3(1) + f^3(2) + ... + f^3(n) + f^3(n+1)= (\frac{f(n+1)f(n+2)}{2})^2 ...(ii)$

$(ii) - (i) \Rightarrow f(n+1) = \frac{f^2(n+2)}{4} - \frac{f^2(n)}{4} = (\frac{f(n+2) + f(n)}{2})(\frac{f(n+2) - f(n)}{2})$

Since, $f(n+1) > 0 \Rightarrow f(n+2) > f(n)$

This , implies that $f(1) < f(2) < ... < f(n)$

Let, $f(n+2) = f(n) + k$

if $k < 2 , f(n+1) < f(n)$ ; contradiction
if $k > 2 , f(n) > f(n+2)$ ; contradiction

Thus , $k = 2 , f(n+1) = f(n) + 1$

$f(1) = 1$.By induction , we can say that $f(n) = n$.

$\therefore f(n) = n$.
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid
Absur Khan Siam
 
Posts: 53
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka


Share with your friends: Facebook Twitter

Return to Algebra

Who is online

Users browsing this forum: No registered users and 1 guest