Please help me with an algebra II question?

Question

My dad and I have been at these problems for 2 hours and just can't figure them out. please help. It's only my second day of school and my teacher gave us this worksheet to work on....

1. Find the sum of the whole numbers 1 to 100, inclusive, without adding them.

- I know it have to do with the nth term or something like that. I supposed to fine the equation and use it for the other problems too. But it's so confusing!!!! >.< Please Help Me!

Answer 1

&quot;The question you asked relates back to a famous mathematician, Gauss.  In elementary school in the late 1700’s, Gauss was asked to find the sum of the numbers from 1 to 100.  The question was assigned as “busy work” by the teacher, but Gauss found the answer rather quickly by discovering a pattern.  His observation was as follows:

    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

Gauss realized then that his final total would be 50(101) = 5050.

The sequence of numbers (1, 2, 3, … , 100) is arithmetic and when we are looking for the sum of a sequence, we call it a series.  Thanks to Gauss, there is a special formula we can use to find the sum of a series:

           s=[n(n+1)]/2

S is the sum of the series and n is the number of terms in the series, in this case, 100.

s= [100(100+1)]/2 = 5050 &quot;