April 10th, 2018, 06:48 PM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 343 Thanks: 1  What is the average of all the odd natural numbers upto 51?
Find the the average of all the odd natural numbers upto 51. Sum of odd numbers is $\displaystyle n^2$. Here n is 51. 51 stands in 26th position in odd natural numbers. Average is $\displaystyle \frac{n^2}{26}$ = 100.03 But answer is 26. Please help me. Last edited by skipjack; April 11th, 2018 at 11:31 AM. 
April 10th, 2018, 06:57 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,250 Thanks: 679  You discovered a great problemsolving tactic earlier. Work hard to solve the problem yourself; since you can't depend on us slackers and mathematical posers. You should challenge yourself to solve this.
Last edited by skipjack; April 11th, 2018 at 11:32 AM. 
April 10th, 2018, 07:03 PM  #3 
Newbie Joined: Nov 2013 Posts: 28 Thanks: 8  1+3+5+..+25+26+...+47 + 49 + 51. Now write the sum as (1+51) + (3+49) + (25+26) = 52*13. So the average will be (52*13)/26 = 26.
Last edited by skipjack; April 11th, 2018 at 11:32 AM. 
April 10th, 2018, 07:24 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 598 Thanks: 366 Math Focus: Dynamical systems, analytic function theory, numerics 
$n$ is not 51. Rethink what the formula $n^2$ is counting. Start by looking at small examples.

April 10th, 2018, 07:28 PM  #5 
Senior Member Joined: Aug 2014 From: India Posts: 343 Thanks: 1  Ok, now can you tell me the procedure how to solve this problem?
Last edited by skipjack; April 11th, 2018 at 11:33 AM. 
April 10th, 2018, 08:35 PM  #6 
Newbie Joined: Nov 2013 Posts: 28 Thanks: 8  1+3+5+..+25+26+...+47 + 49 +51. Now write the sum as (1+51) + (3+49) + ...+(25+26) = 52*13. So the average will be (52*13)/26 = 26.
Last edited by skipjack; April 11th, 2018 at 11:34 AM. 
April 10th, 2018, 08:41 PM  #7 
Senior Member Joined: Aug 2014 From: India Posts: 343 Thanks: 1  Why did you write (1+51) + (49 +3)..... etc?
Last edited by skipjack; April 11th, 2018 at 11:35 AM. 
April 10th, 2018, 09:21 PM  #8 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 2,001 Thanks: 132 Math Focus: Trigonometry  That's basically what Gauss did to invent the arithmetic summation formula.
Last edited by skipjack; April 11th, 2018 at 11:36 AM. 
April 10th, 2018, 10:02 PM  #9 
Senior Member Joined: Aug 2014 From: India Posts: 343 Thanks: 1  If n is not 51. Then n is 26. Can you tell me why n is not 51?
Last edited by skipjack; April 11th, 2018 at 11:35 AM. 
April 11th, 2018, 10:14 AM  #10 
Newbie Joined: Nov 2013 Posts: 28 Thanks: 8  In order to get the same sum in each parenthesis. If you keep adding 52 to itself say 11 times, the sum will be 52*11.
Last edited by skipjack; April 11th, 2018 at 11:36 AM. 

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