Question

Use Gauss's approach to find the following sums do not use formulas.
a.) 1+2+3+4+.....+1002
b.) 1+3+5+7+.....+997

Answer 1

but Gauss's approach to find the sums requires the Gauss formula :)

Answer 2

A formula young Gauss discovered when trying to add the numbers 1-100, which goes n(n+1)/2, which is (50*51)/2, or 5050. He found it by discovering that 100+1=101, 99+2=101, 3+98=101...50+51=101. He found out that by wrapping the numbers around, that is adding (100-1) and (1-100) together, he would get the number 100 at every term. He then multiplied 100 times the number of terms, and got his answer, when dividing it by 2, because he had added twice the sum he wanted. Hence the sum of all integers between 1 and 100 is 101*50=5050.

Answer 3

http://www.coolmath.com/algebra/19-sequences-series/06-gauss-problem-arithmetic-series-01.htm