I am not sure how to solve this GRE question

Question

k = 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 Which is greater, k or 2?

Answer 1

Suppose you want to walk 1 mile.

1st you walk 1/2 of 1 mile, ie you walk 1/2 mile. And 1/2 mile remains.

Then you walk 1/2 of 1/2 =1/4. Then 1/4 mile remains.

Then you walk (1/2) of (1/4) which is an 1/8. Then 1/8 remains.

If you continue you get that 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +1/64 + 1/128 + 1/256 + 1/512 +....

So just part of that will be less than 1

That is 1 + (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128) < 2

Answer 2

K is a geometric infinite series with common ratio 1/2.

Yes the total converges to 2, specifically 1 / (1-1/2) = 2

Since K is defined as a FINITE sequence, the sum shall be LESS than 2.

k<2