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More ways to get Braingle...

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?

Open Calculator

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

mitzimesserAil*
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*en*
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...
MarcM1098Aca*
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*us*
Mar 05, 2006

WOW, I THINK
stil*us*
May 12, 2006

"Explainers"-
Please replace "between 1 and 100" with "between 0 and 100" or "betweem 1 and 99, inclusive." Keep it consistant.
calmsaviorAus**
Sep 22, 2006

why did i find this Easy?
calmsaviorAus**
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*us*
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.
PaladinAus*
Mar 13, 2009

the sum of the first n odd numbers is n^2
PaladinAus*
Mar 13, 2009

Ok, my comment looks really lame after reading the others but yeah, the answer is correct
doeheadAus*
Jun 25, 2009

My experience as an actuary made this quite easy. But I'm sure it was hard for most people.
kwelchansAus*
Jun 25, 2009

I figured out this trick in college (I was a math major) so this was easy for me. Good teaser!
triskelexAus
Jun 25, 2009

And for the sum of even numbers its ( N x N ) + N
auntiesisAus*
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_ltus*
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!
4princessesus*
Jun 25, 2009

huh? totally don't get it!
Mill89Aus*
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*us*
Jun 25, 2009

sbminerb is the first comment that makes sense to me. Thanks!
bradon182001*us*
Jun 29, 2009

Interesting.
alpha_zero924Aus*
Jul 05, 2009

Also, if you wanted to do even numbers it would be n^2 + n
TZCastortus
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.
dalfamnestAnz*
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*us*
Mar 09, 2011

vintagepotato: read my explanation above if you want to understand why your solution also works.
Candi7tus*
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!!!
iggy39Aus*
Jun 25, 2012

boring sorry not a math person.



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