For 1st, check this out
https://www.beatthegmat.com/gmat-prep-qu ... tml#206219

For 2nd,
let's plugin values for each choice and see which choice satisfies f(a+b)=f(a)+f(b).

A. Assume f(a)=a^2, f(b)=b^2. f(a+b)=(a+b)^2 (RHS), which is not equal to a^2 + b^2 [f(a) + f(b)]
B. Assume f(a)=a+1, f(b)=b+1. f(a+b)=a+b+1 (RHS), which is not equal to (a+1)+(b+1) = a+b+2 [f(a) + f(b)]
C. Assume f(a)= -3a, f(b)= -3b. f(a+b)= -3(a+b) = (-3a) + (-3b) which is equal to f(a) + f(b). This choice satisfies.

By this time, you can just end this question by ticking C. But i'm just showing you other 2 choices as well.

D. Assume f(a)= sqrt(a), f(b)=sqrt(b). f(a+b) = sqrt(a+b) (RHS), which is not equal to sqrt(a)+sqrt(b) [f(a) + f(b)]
E. Assume f(a)=2/3, f(b) = 2/3. f(a+b)=2/3 (RHS), which is not equal to 2/3 + 2/3 [f(a) + f(b)]

So, its C.

thank you so much...dang, you re very smart!!!

K=1 => 1/2
K=2 => -1/4
K=3 => 1/8
K=4 => -1/16

One way is to use GP and solve it, other one which I found time saving was

1/2 - 1/4 + 1/8 - 1/16 ...

first 2 terms 1/2 - 1/4 = 1/4
Rest terms 1/8 - 1/16 .... < 1/4

Hence answer is between 1/4 & 1/2, i.e. D