applying number theory

Question

3) Find the sum of the even integers between 30 and 70.

Answer 1

32+34+36+...+64+66+68=?  (there are 19 numbers: 68-32=36, 36/2=18, 18+1=19, or just count them)

S(19)=(19/2)(2*32+[19-1]2) because this is an arithmetic sequence

S(19)=(19/2)(64+36)

S(19)=(19/2)(100)

S(19)=19*50

S(19)=950

or,....

32+68=100

34+66=100

36+64=100

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exclude 50 which is the middle number

48+52=100

there are 9 groups of 100 plus the 50 in the middle which equals 900+50=950