Question

1.what is the sum of each of the pairs 1 and 100,2 and 99,3 and 98.....50 and 51?

2.how many pairs are there in #1?

3.From your answer in #1 and #2,how do you get the sum of integers from 1 to 100?

4.what is the sum of the integers from 1 to 100?​

Answer 1

Answer:

1) 1+100 is 101

2+99 is 101

3+98 is 101

And so on...

There are 50 of these pairs and the total sum of integers from 1 to 100 is 5050

Answer 2

Answer:

1....For the sum of 1 to 100, that would be using the arithmetic series formula

Sn = n/2 (a1 + an)

Sn = 100/2 ( 1 + 100)

Sn = 50(101)

Sn = 5050

2...null

3...1 + 2 + 3 + 4 + … + 98 + 99 + 100

Gauss noticed that if he was to split the numbers into two groups (1 to 50 and 51 to 100), he could add them together vertically to get a sum of 101.

1 + 2 + 3 + 4 + 5 + … + 48 + 49 + 50

100 + 99 + 98 + 97 + 96 + … + 53 + 52 + 51

1 + 100 = 101

2 + 99 = 101

3 + 98 = 101

.

.

.

48 + 53 = 101

49 + 52 = 101

50 + 51 = 101

1 + 2 + 3 + ... + 19 + 20 + 21

21 + 20 + 19 + ... + 3 + 2 + 1

Now if you add vertically you get

22 + 22 + 22 + ... + 22 + 22 + 22 = 21(22) = 462

But this is twice the sum of the first 21 whole numbers so

1 + 2 + 3 + ... + 19 + 20 + 21 = 462/2 = 231

4...null