Question

When Gauss was very young, a teacher told him to find the sum of all the numbers from 1-100, thinking this would keep him busy. Gauss cleverly paired the numbers and observed that 1+99=100, 2+98=100, 3+97=100, and so on.

Answer 1

How many pairs of numbers can be formed this way?


shoe me how you got it too pleaseeeeeeeeeeee

Answer 2

50

Answer 3

really? how

Answer 4

100/2 = 50 pairs

Answer 5

err.. lol 100 pairs ... im getting aheadf of myself

Answer 6

are you sure? there's an answer sheet and it says 49 pairs..

Answer 7

do you know they got it?

Answer 8

lets start small if need be:

1,2,3,4,5,6 ; how many pairs can you make similar to the above question?

Answer 9

6,1
5,2
4,3
3,4
2,5
1,6

6 pairs total

Answer 10

but you may notice that you have counted twice as many times as need be when you go that route; so we have to divide it by half

Answer 11

what i did was
99,1
98,2
97,3
......

Answer 12

guass observed that the first and last numbers added; not the first and next to last numbers

Answer 13

..?

Answer 14

isn't what you just said the same??

Answer 15

it still works with an odd count to begin with; but only cause the math is forgiving :)

Answer 16

100+1 = 101
99+2 = 101
98+3 = 101
......

Answer 17

??

Answer 18

101, 100 times = 10100; but youve counted each set twiice, so divide in half to get 5050

Answer 19

how'd you get that

Answer 20

the same way i got it with the smaller example

Answer 21

n*(first+last)/2

Answer 22

n=??

Answer 23

i spose if you start counting with 99+1, itll be 49; but that just seems wierd

Answer 24

what does n equal?

Answer 25

n = number of terms in the set

Answer 26

1,2,3,...,98,99,100 ; has 100 terms in the set right?

Answer 27

yea

Answer 28

so...100*(first+last)/2 ??

Answer 29

so you count 100 pairs of (101);

100+1 = 101
99+2 = 101
98+3 = 101
-1+1 = 101

Answer 30

100(100+1) = 10100

now tell me, does 1+100; and 100+1 count as seperate or the same?

Answer 31

OH i understand that, but how did the answer book get 49 for the pairs of numbers?

Answer 32

it's the same, right

Answer 33

??

Answer 34

the started counting off at 99+1 instead of doing it sensibly

Answer 35

so how would i show work for that? would i have to write down the numbers 1-100??

Answer 36

the sum of all the numbers from 1-99 is 99(99+1)/2; which is not the same

Answer 37

if your teacher wants you to show all the work instead of just an understanding of it, then yes; youd write them all out

Answer 38

...aw

Answer 39

the thing is you shouldnt have to start at 100 to understand the concept ...

Answer 40

its the same concept for 2 numbers as it is for 100 or 100000

Answer 41

it even works for 1 number :)

Answer 42

whaaa..?

Answer 43

what is the sum of 1?

Answer 44

umm 1?

Answer 45

yep; so if we count it twice; what do we get? 1+1 = ?

Answer 46

2..

Answer 47

what does this all mean??

Answer 48

and how do we get it back to its proper sum? divide by 2 right?

Answer 49

lets try it with 2 numbers,

1,2 ; what is the sum of the numbers?

Answer 50

3

Answer 51

now what do we get when we sum up pairs?

2+1 = 3
1+2 = 3
--------
6 right?

Answer 52

yea

Answer 53

but you can see that we simply counted twice as much as we needed to right?

Answer 54

yea

Answer 55

but you're not suppose to count twice

Answer 56

wait is that why you divided by 2?

Answer 57

Yes :)

Answer 58

ok

Answer 59

by counting pairs, we actually end up counting it all 2 times; so we have to adjust for that

Answer 60

so how would i show my work? the sum of the numbers 1-100 is 5050

Answer 61

i don't want to take up too much time and space, so what would be the easiest way?

Answer 62

you would either have to show that all the pairs of numbers from 1 to 100 is double the actual sum; or, devise a plan from recursive equations

Answer 63

..??

Answer 64

without some basic background in recurrsives; the long way is about the only way i can see to prove it

Answer 65

oh

Answer 66

it either that or prove it for the smaller cases and then generalize it for larger cases

Answer 67

how would i show that all the pairs of numbers from 1-100 is double the actual sum though

Answer 68

how did you get 10100 anyways?

Answer 69

Show that 1+100 = 100+1 id say.

Answer 70

101*100 = 10100

Answer 71

i counted up the 100 pairs that summed to 101

Answer 72

ok

Answer 73

you want to know how it is made 100*(101)/2

Answer 74

uhh??

Answer 75

1 + 2 + 3 + ... + 98 + 99 + 100
100 + 99 + 98 + ... + 3 + 2 + 1
----------------------------------
101+101+101 + ... +101+101+ 101

Answer 76

ok i understand now

Answer 77

if i don't understand can i call u in the chat..?

Answer 78

i dont go to chat alot, jamal might tho

Answer 79

who..? nvm thanks :D

Answer 80

OHH jamal..

Answer 81

I want to tell you another method that is very powerfull by which you can know sum of squares, cubes what so ever

Answer 82

ok

Answer 83

x^2-(x-1)^2=2x-1
1^2-(0)^2=2(1)-1
2^2_1^2=2(2)-1
.....................
(n-1)^2-(n-2)^2=2(n-1)-1
n^2-(n-1)^2=2n-1
adding all the terms
you will see lot of terms are delting
n^2=2(1+2+3....n)-(1+1+1+1.....n)
n^2=2(1+2+3....n)-n
so 1+2+3...n=n(n+1)/2