The number of marbles in each set is an arithmetic sequence. The sum of any arithmetic sequence is that average of the first and last terms times the number of terms...mathematically this is:
s(n)=(2an+dn^2-dn)/2, a=initial term, n=term number, d=common difference
In this case the first term is 2 and since each term is 3 more than the previous term, the common difference is 3, so now we can say:
s(n)=(2*2n+3n^2-3n)/2
s(n)=(3n^2+n)/2 and we are told that there is a total of 40 so
(3n^2+n)/2=40
3n^2+n=80
3n^2+n-80=0
3n^2-15n+16n-80=0
3n(n-5)+16(n-5)=0
(3n+16)(n-5)=0, since n>0
n=5
So Abe will make five sets of marbles...
(The sets contain 2,5,8,11, and 14 marbles, which sum to 40 marbles total)
