a. Let S be the first sum,

S = 1 + 2 + 3 + … + 97 + 98 + 99

If we reverse the order of terms, the value of the sum is unchanged:

S = 99 + 98 + 97 + … + 3 + 2 + 1

If we add up the terms in both version of S in the same positions, we end up adding 99 copies of quantities that sum to 100 :

S + S = (1 + 99) + (2 + 98) + … + (98 + 2) + (99 + 1)

2S = 100 + 100 + … + 100 + 100

2S = 99 × 100

S = (99 × 100)/2

Then S has a value of

S = 99 × 50

S = 4950

Aside: Suppose we had n terms in the sum, where n is some arbitrary positive integer. Call this sum ∑(n) (capital sigma). If ∑ is a sum of n terms, and we do the same manipulation as above, we would end up with

2 ∑(n) = n × (n + 1) ⇒ ∑(n) = n (n + 1)/2

b. Let S' be the second sum. It looks a lot like S, but the even numbers are missing. Let's put them back, but also include their negatives so the value of S' is unchanged. In doing so, we have

S' = 1 + 3 + 5 + … + 1001

S' = (1 + 2 + 3 + 4 + 5 + … + 1000 + 1001) - (2 + 4 + … + 1000)

The first group of terms is exactly the sum ∑(1001). Each term in the second grouped sum has a common factor of 2, which we can pull out to get

2 (1 + 2 + … + 500)

so this other group is also a function of ∑(500), and so

S' = ∑(10001) - 2 ∑(500) = 251,001

However, we want to use Gauss' method. We have a sum of the first 501 odd integers. (How do we know there 501? Starting with k = 1, any odd integer can be written as 2k - 1. Solve for k such that 2k - 1 = 1001.)

S' = 1 + 3 + 5 + … + 997 + 999 + 1001

S' = 1001 + 999 + 997 + … + 5 + 3 + 1

2S' = 501 × 1002

S' = 251,001

c/d. I think I've demonstrated enough of Gauss' approach for you to fill in the blanks yourself. To confirm the values you find, you should have

3 + 6 + 9 + … + 300 = 3 (1 + 2 + 3 + … + 100) = 3 ∑(100) = 15,150

and

4 + 8 + 12 + … + 400 = 4 (1 + 2 + 3 + … + 100) = 4 ∑(100) = 20,200