Question

Legend has it that the great mathematician Carl Friedrich Gauss(1777-1855) at a very young age was told by his teacher to find the sum of the first 100 counting numbers. While his classmates toiled at the problem, Carl simply wrote down a single number and handed the correct number to his teacher. The young Carl explained that he observed that there were 50 pairs of numbers that each added up to 101. So the sum of all the numbers must be 50*101=5050. 1+2+3+.....+355=

Answer 1

63190

Explanation:

Given that the sum of the first 100 counting numbers can be divided into 50 pairs, each with a sum of 101, you want the sum of the first 355 counting numbers.

Counting pairs

The numbers 1 to 100 can be paired, because there is an even number of them.

The numbers 1 to 355 can be similarly paired to form 127 pairs totaling 1+355=356. The middle pair of the list will be 177 +179 = 356. That is, the even number 178 is not paired with any other number in the list.

This makes the total of numbers 1 to 355 be ...

177×356 +178 = 63,190

Alternatively, we can pair the numbers 1 to 354 into 177 pairs totaling 355, then add the final number, 355 to the sum:

177×355 +355 = 178×355 = 63,190

The total 1+2+3+...+355 is 63,190.

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Additional comment

The formula applicable to this sum is ...

S = n(n+1)/2

For even n, this is (n/2)(n+1), the sum that Gauss used for 1..100. For odd n, the sum is (n+1)/2×n, the alternate sum we used above.