What is the Pattern for Cumulative Sums in a Summations Type Problem?

[Jamezy]

Homework Statement

Write down the numbers 1,2,3, ….
Delete every third number, beginning
with the third. Write down the
cumulative sums of the numbers which
remain. That is:
1 2 3 4 5 6 7 …
1 2 4 5 7 …
1 3 7 12 19 …
Now delete every second number,
starting with the second, and write
down the cumulative sums of what
remains


I know that it always ends up as the cubic numbers ie:

1 8 27 64 etc

But how would I make a proof of this?

Homework Equations

Summations of r, r^2 and 1 between 1 and n



Literally don'tknow how to do it at all!

 

[Dick]

Try working forwards and backwards from the sequence just before you did the last cumulative sum yielding the cubes. That was 1,7,19,37,61,... Look at difference between successive elements. That gives you 6,12,18,24,... there's a pretty obvious pattern there. Can you work forward to show the successive sums of that are the cubes? Now can you look back and see how to prove how those differences come about?