Braingle: 'Number Tricks' Brain Teaser

Number Tricks

Math brain teasers require computations to solve.

Puzzle ID:	#28646
Fun:	(2.93)
Difficulty:	(2.04)
Category:	Math
Submitted By:	____
Corrected By:	MarcM1098

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?

Hint

Consecutive odd numbers are 1, 3, 5, 7....
Start with the first several odd numbers and look for a pattern.
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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 = (x-1)^2 + x + (x-1) = x^2 - 2x + 1 + x + (x-1) = 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.
Metaphor 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*(j-i+1)
lookagain 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.
vintagepotato Nov 19, 2010	For some reason I found the answer by taking (n-1)(n+1)+1. (50-1)(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.