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?

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

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...

Show that all pentagonal numbers are one third of a triangular number.

These alphabet bricks are painted in a special way. A is on one brick, B on two bricks, and so on. How many bricks will be painted by the time they have got to other letters of the alphabet?