7^1 = 7

7^2 = ..9

7^3 = …3

7^4 = …1

7^5 = …7

So you see that it repeats after 4 terms. This means that all powers that are multiples of 4 will give you one 1, multiples of 4 plus 1 gives 7, and so on. 34 is 32 + 2, so it is a multiple of 4 plus 2, which means that the units digit will be 9.

Next one!

Well, 0 doesn’t add anything, so let’s just start with 2. Let’s say you write down half the list on one side, and the other half on the other. (Don’t actually do that. It’s too long.) Like so:

2 398

4 396

6 394

8 392

ETC. You can see that each pair will add up to 400. The middle number is 200 and it won’t be paired with anything. So the number of pairs is the number of even numbers from 2-198 inclusive, which means it’s (198-2)/2 + 1. You have 98 pairs.

The sum is 98*400 + 200.