Please do not post more than two problems per thread. One problem is preferred. I recommend opening new threads for problems 4) — 7). (By the way, where is problem #3?)

I do not recognize the pattern in 1 × 1513 + 2 × 1512 + 3 × 1511 + ….+ 13 × 15. It seems that the kth term is k × (1500 + 14 - k). But then the last term should be 13 × 1501, not 13 × 15.

Does n(A × B) denote the number of elements in A x B? In any case, A and B are not well-defined. You cannot write a condition like 1/a + 1/b = 1/14 outside the curly braces in the set-builder notation. Either A = {a: a is an integer}, period, and then 1/a + 1/b = 1/14 is not allowed because the definition of A is finished and it is not clear whatais anymore, or A = {a: a is an integer and 1/a + 1/b = 1/14}, which does not work either because it is not clear whatbis.