Number Tricks
Math brain teasers require computations to solve.
Ms. Arroyo asked the class to see if they could find the sum of the first 50 odd numbers. As everyone settled down to their addition, Terry ran to her and said, "The sum is 2,500." Ms. Arroyo thought, "Lucky guess," and gave him the task of finding the sum of the first 75 odd numbers. Within 20 seconds, Terry was back with the correct answer of 5,625.
How does Terry find the sum so quickly?
HintConsecutive odd numbers are 1, 3, 5, 7....
Start with the first several odd numbers and look for a pattern.
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Answer
The following pattern holds: The sum is equal to n x n, when n is the number of consecutive odd numbers, starting with 1. For example, the sum of the first 3 odd numbers is equal to 3 x 3, or 9; the sum of the first 4 odd numbers is equal to 4 x 4, or 16; the sum of the first 5 odd numbers is equal to 5 x 5, or 25; and so on.
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Comments
mitzimesser
Feb 15, 2006
 I don't get it, 50*50=2500
I don't understand it. 
miyukina
Feb 15, 2006
 i think you worded it wrong...it's not 'the first 50 odd numbers', coz that'll mean [1,3,5...98,99]...should be 'the sum of all odd numbers up to 50' if you wish the answer to be 625, else it'd be 2500 instead... 
Chakoteya
Feb 15, 2006
 Your explanation makes your example wrong....
Tut. 
blackmarket69
Feb 19, 2006
 ur wrong n da teaser. 50x50=2500 
mikey38
Feb 24, 2006
 The "sum of all the odd numbers up to 75" is 1444. 
miyukina
Feb 26, 2006
 morning rain8, i think you corrected some correct part, too... 
MarcM1098
Feb 28, 2006
 morning_rain08, the puzzle is wrong. Use either of the following:
1.
The sum of the first 50 odd numbers (those between 1 and 100) is 50x50=2500.
The sum of the first 75 odd numbers (those between 1 and 150) is 75x75=5625.
2.
The sum of all odd numbers up to 50 (those between 1 and 50) is 25x25=625.
The sum of all odd numbers up to 75 (those between 1 and 75) is 38x38=1444. 
solarsistim321
Mar 05, 2006
 WOW, I THINK 
stil
May 12, 2006
 "Explainers"
Please replace "between 1 and 100" with "between 0 and 100" or "betweem 1 and 99, inclusive." Keep it consistant. 
calmsavior
Sep 22, 2006
 why did i find this Easy? 
calmsavior
Sep 22, 2006
 75x75=5625
any number ending in 5 can easily be squared.
just take the number adjacent to the last 5:
75 => 7
145 => 14
35 => 3
then you multiply it with the number just one above it:
7x8= 56
then add 25 to the end
56 25 => 5625!
(exclamation point not factorial) 
javaguru
Dec 29, 2008
 The reason this works is because
x^2 = (x1)^2 + x + (x1)
= x^2  2x + 1 + x + (x1)
= x^2
this means that the difference between x^2 and (x+1)^2 is 2x + 1 (e.g. 10^2 = 100, 11^2 = 121, 121  100 = 21 = 10 + 11.)
1^2  0^2 = 1
2^2  1^2 = 3
3^2  2^2 = 5
4^2  3^2 = 7
. . .
So summing the first n odd numbers gives n^2. 
Paladin
Mar 13, 2009
 the sum of the first n odd numbers is n^2 
Paladin
Mar 13, 2009
 Ok, my comment looks really lame after reading the others but yeah, the answer is correct 
doehead
Jun 25, 2009
 My experience as an actuary made this quite easy. But I'm sure it was hard for most people. 
kwelchans
Jun 25, 2009
 I figured out this trick in college (I was a math major) so this was easy for me. Good teaser! 
triskelex
Jun 25, 2009
 And for the sum of even numbers its ( N x N ) + N 
auntiesis
Jun 25, 2009
 I love math, but never knew this trick. Very cool ! 
(user deleted)
Jun 25, 2009
 1st 50 odd numbers are:
1, 3, ..., 97, 99
Add 1+99=100, 3+97=100, ..., 49+51=100 (25 times).
25*100=2500 
tova_l
Jun 25, 2009
 wow this teaser brings lots of comments...nothing to say, it was pretty hard for me . but once i saw the answer, it amazed me that math is so precise! 
4princesses
Jun 25, 2009
 huh? totally don't get it! 
Mill89
Jun 25, 2009
 The early comments are confusing, since it must have been worded incorrectly at first.
I, too, am a bit of a math nerd but never knew this before, so it's a neat trick for me. 
jcann
Jun 25, 2009
 sbminerb is the first comment that makes sense to me. Thanks! 
bradon182001
Jun 29, 2009
 Interesting. 
alpha_zero924
Jul 05, 2009
 Also, if you wanted to do even numbers it would be n^2 + n 
TZCastor
Oct 05, 2009
 Brilliant, i got it with my first try. For the guys that are confised, it's the first 50 ODD numbers, not just the first 50 numbers. So in conclusion, the range is to a hundred. 
dalfamnest
Oct 13, 2009
 I guess this must have been corrected  no problem there now, the way I see it.
I used a 'series' method of finding the sum. The numbers are 1+3+5+ ...+97+99. Adding 1+99, 3+97, 5+95 etc gives us 25 pairs of numbers whose sum is 100. 25x100=2500. 
(user deleted)
Oct 08, 2010
 I used the summation technique:
Let Sij[pattern] = "Summation from i to j of pattern"
First 50 odd numbers = S0,49[2n+1]
= 2S0,49[n] + S0,49[1]
= 2x[49(49+1)/2] + 50
= 49(50) + 50
= 50(49 + 1)
= 50(50), 2500, or n^2
**note that Sij[n] refers to the arithmetic sequence
**also note that Sij[CONSTANT NUMBER] = Constant*(ji+1) 
(user deleted)
Nov 08, 2010
 There is no sum of the first 50 odd numbers, nor is there a sum of the first 75 odd numbers, BECAUSE
odd numbers are also negative.
Rephrase part of the question with
"the sum of the first 50 odd POSITIVE numbers," etc. 
(user deleted)
Nov 19, 2010
 For some reason I found the answer by taking (n1)(n+1)+1.
(501)(50+1)+1
49*51+1
2499+1
2500
I have no idea how I came to this conclusion, but what the hey.. It works, doesn't it? 
javaguru
Mar 09, 2011
 vintagepotato: read my explanation above if you want to understand why your solution also works. 
Candi7
Jun 15, 2012
 COOLNESS!!! I used to use that all the time, but then I forgot about it. Now I realize.... it's always a square! I also know a handy technique for squaring ANY number ending in 5 in a matter of seconds. 
Babe
Jun 25, 2012
 Good grief! Another math problem to solve. These are not even fun. At least not to me. Don't suggest I not even try it as I DID NOT!!! 
iggy39
Jun 25, 2012
 boring sorry not a math person. 
Zykezex
Jun 25, 2015
 It can be solved in seconds by opening an Excel Spreadsheet and dragging down the first 50 & 75 odd numbers and hitting autosum LOL! 
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