Adding Consecutive Numbers

1
Adding Consecutive Numbers

• A day in the life of
• Carl Guass

2
This is a true story!
3
The date was 1787in Braunschweig,Germany.
4
(No Transcript)
5
Carl Gauss was 10 years old.
6
(No Transcript)
7
Carls math teacher angrily assignedthe class
of boys atedious problem.
8
Ill make you rotten kids pay for your behavior.
9
Their problem was to sum all of the numbers
from 1 to 100.
10
Boring!!!
11
After only a fewseconds, young Carlwas finished
withthe problem.
12
Ive got the answer!Its 5050.
13
His teacher wasshocked.
14
I dont believe it. How in the world did you get
this answer so quickly?
15
Its actually quite easy.
16
Lets consider asimpler problem.
17
Lets sum all of thenumbers from1 to 10.
18
First divide thelist of numbersinto 2 equal
columns.
19
12345
678910
20
Sum pairs of numbers in the following manner
21
Sum the first and the last .
12345
678910
1 10 11
22
Sum the second and the second to last .
12345
678910
2 9 11
23
Sum the third and the third to last .
12345
678910
3 8 11
24
Sum the fourth and the fourth to last .
12345
678910
4 7 11
25
Sum the fifth and the fifth to last .
12345
678910
5 6 11
26
When the numbers are paired like this...
12345
678910
their sum is always 11.
27
5 sets of 11 make 55.
12345
678910
28
Lets try to sumall of the numbersfrom 1 to 20.
29
Begin by dividingthe list of numbers into 2
groups.
30
12345678910
11121314151617181920
31
Now start to pair-up numbers.First to
last,second to second to last,etc.
32
Every pair equals 21
12345678910
11121314151617181920
33
Every pair equals 21
12345678910
11121314151617181920
There are 10 sets of 21
34
10 x 21 210
35
Now, lets do all of the numbers from 1 to 100.
36
As always, beginby dividing the list of
numbersinto 2 equal columns.
37
12345...4950
515253...979899100
38
Pair-up the numbers.
39
Each pair of numbers sums to 101
12345...4950
515253...979899100
40
Each pair of numbers sums to 101
12345...4950
515253...979899100
There are 50 sets of 101
41
50 x 101 5050
42
Now its time for you to try.
43
Sum all of the numbers from 1 to 40.
44
Every pair of numbers equals 41
12345...1920
2122232425...3940
45
There are 20 sets of 41
12345...1920
2122232425...3940
20 x 41 820
46
Sum all of the numbers from 1 to 50.
47
Every pair of numbers equals 51
12345...2425
2627282930...4950
48
There are 25 sets of 51
12345...2425
2627282930...4950
25 x 51 1275
49
Lets sum all of the the numbers from 1 to 35.
50
This problem is a little different because
there isan odd number ofnumbers.
51
Lets pretend thatwe are summing allof the
numbers from1 to 34, and then well add on 35.
52
Every pair of numbers equals 35
12345...1617
1819202122...3334
53
There are 17 sets of 35
12345...1617
1819202122...3334
17 x 35 595
54
Dont forget to addthe extra 35 that we left
off.
55
595 35 630
56
This strategy of summing consecutive numbers is
used a lotin the field of probability
andstatistics.
57
The End.