Before trying the problems "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Part A. Write down the positive integers, cross out every second, and form the partial sums of the remaining.

1 2 3 4 5 6 7 8 9 10 11
1   4   9   16  25   36

Does the pattern of squares continue?

Part B. As before, but cross out every third, form partial sums, then cross out every second and for a second partial sums.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 3 7 12 19 27 37 48 61 75 91
1 8 27 64 125 216

Does the pattern of cubes continue?

The first one is pretty familiar.

Some tinkering with the second is needed:

The first set of partial sums is: sum 1 to n (6n)+1, which is 3n^2+3n+1.

Then we are taking partial sums again:
sum 1 to n (3n^2+3n+1), which is  (n+1)^3 - 1+1 = (n+1)^3.

So yes, it will always be a cube.

I got this one right.

Edited on September 25, 2017, 1:56 pm

Posted by broll on 2017-09-25 13:51:34