Carcass wrote:
What is the value of \(2 + 2^1 + 2^2 + 2^3...........2^{18}\)?
A) \(2^{19}\)
B) \(2^{171}\)
C) \(2^{172}\)
D) \(2^{18!}\)
E) \(2 + 2^{18!}\)
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITIONGRE - Math BookOne approach is to start listing values and look for a pattern (I'll let others do that)
We can also solve the question quickly by
eliminating 4 of the answer choices.
Notice what would happen if we took the sum 2 + 2^1 + 2^2 + 2^3 + . . . . 2^17 + 2^18 replaced each value with 2^18
The
NEW sum would definitely be bigger than the
ORIGINAL sum
That is:
2 + 2^1 + 2^2 + 2^3 + . . . . 2^17 + 2^18 <
2^18 + 2^18 + 2^18 + 2^18 . . . + 2^18 + 2^18Notice that the
NEW sum is the sum of
nineteen 2^18's
So, we can write:
2 + 2^1 + 2^2 + 2^3 + . . . . 2^17 + 2^18 < (
19 )
(2^18)Now notice that
19 <
2^5, so we can write:
2 + 2^1 + 2^2 + 2^3 + . . . . 2^17 + 2^18 < (
19 )
(2^18) < (
2^4 )
(2^18)Simplify the right-most expression to get:
2 + 2^1 + 2^2 + 2^3 + . . . . 2^17 + 2^18 < (
19 )
(2^18) <
2^22So, we can conclude that:
2 + 2^1 + 2^2 + 2^3 + . . . . 2^17 + 2^18 <
2^22Only answer choice A is less than
2^22