Need to think a bit about part a), but for now, here is some help with part b) if you set x = 2 / k (then note that x goes to zero as k goes to infinity.

The typical term of the sequence becomes A_k = (1 + x) ^(2/x) =

= [(1+x)^(1/x)]^2 whose limit, as x goes to zero, is e^2, where e is Euler's Constant. Some numerical investigation in Excel also verifies:

k A_k

1 3

2 4

3 4.62962963

4 5.0625

5 5.37824

6 5.618655693

7 5.8077951

10 6.191736422

15 6.536796047

100 7.244646118

1000 7.37431239

10000 7.387578632

Good luck!!