Question

Use gauss approach to find the following sum (do not use formulas)
A. 1+3+5+7+...+101

Answer 1

there is a simple formula for both of these, but i think "gauss approach" means add forward and backward, then divide by 2

\[S=1 + 2+ 3+...+1000+ 1001+1002\]
\[S=1002+1001+1000+...+ 3 + 2 +1\]
\[2S=1003+1003+1003+...+1003\]
\[2S=1002\times 1003\]
\[S=\frac{1002\times 1003}{2}\]

Answer 2

gauss's approach is to reverse the order then add the vertical pair

e.g

1 2 3 4 5 6
+ 6 5 4 3 2 1
---------------------
7 7 7 7 7 7

the vertical pairs will always add to the same value

to you have 6 lots of 7 so 6 x 7 = 42

and becuase you have 2 lots of the numbers 1 to 6 you need to halve the answer

42/2 = 21 so the sum of the 1st 6 numbers is 21

Answer 3

from this derive the formula
\[\sum_{k=1}^nk=\frac{n(n+1)}{2}\]

Answer 4

so in your question for the sum of the numbers

1 2 3 4 ........ 1002
+1002 1001 1000 999 1
------------------------------------
1003 1003 1003 1003 1003

you have 1002 lots of 1003 then halve that answer