Triangle numbers

Here is a collection of puzzles about Sam's shop sent in by club members. Perhaps you can make up more puzzles, find formulas or find general methods.

Can you find a rule which connects consecutive triangular numbers?

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

Can you find a rule which relates triangular numbers to square numbers?

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...