Does anyone know an easy way to add all numbers from 1 - 100?

Question

Leonhard Euler did this as a 3rd grader in a matter of about 2 minutes. He added 1+ 100 = 101. I am very interested in the method than the number.

Answer 1

Look at it like this:

100 + 1 = 101
99 + 2 = 101
98 + 3 = 101
97 + 4 = 101
...

So, if you add all of the 101's up, you have 100 of them, or 100 * 101. But you added 100 twice, 99 twice, 98 twice, ... 2 twice and 1 twice. So, you divide by 2 to get the final answer:

100 * 101 / 2 = 5050

Answer 2

The method is very nice. You need think of it this way:

1 + 2 + 3 + 4 + 5 + 6 + ... + 100
100 + ... + 6 + 5 + 4 + 3 + 2 + 1
========================
101 + 101 + 101 + 101 + ... + 101 (the 101 comes from adding 1 + 100, 2 + 99, 3 + 98, etc.)

So it is the same as 100*101, but then you have two copies of it.

So the answer is (100*101)/2

Answer 3

Write out 1 + 2 + 3 + ... + 99 + 100 and under it
write out 100 + 99 + 98 ... + 2 + 1, then add vertically.

You get 101 + 101 + ... + 101, one hundred terms. So the total of all that is 100(101) = 10100. But this is the sum you wanted, twice, so divide by 2 and get 5050.

Notice that 101 is the sum of the 1st term and the last, and that 100/2 is half the number of terms. So to add all the integers from 10 to 1000, which is 990 terms, compute 990(1010)/2.

Answer 4

the formula is n*(n+1)/2 where n=100 in this case
100*101/2=5050

Answer 5

my professor showed me how to do this the way he did but I can't remember)However the only way i can think of is summation. it is 100*101 and then that divided by 2. you take {n(n+1)}/2 so it would be 5050