What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?

Question

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!}

pushpitkc · Accepted Answer

2 + 2^1 = 2 + 2 = 4 = 2^2

Similarly,  2 + 2^1 + 2^2 = 2^3  and  2 + 2^1 + 2^2 + 2^3 = 2^4  and so on..

Now,  2 + 2^1 + 2^2 + 2^3 + ...... + 2^{17} = 2^{18}

Therefore, the value of the expression  2 + 2^1 ........... + 2^{17} + 2^{18}  =  2^{18} + 2^{18}  =  2^{18} * 2  or  2^{19}  (Option A)

niks18 · Answer

If we use the concept of Progression (specifically Geometric Progression), then the question can be solved very easily. The series can be arranged as -

2 + { 2^1 + 2^2 + 2^3...........2^{18} }. Numbers within the bracket is a GP with  first term(a) = 2 ,  common ratio(r) = 2  and  number of terms(n) = 18

Sum of GP =  a*\frac{(r^n -1)}{(r-1)}  =  2*\frac{(2^{18}-1)}{(2-1)}  =  2^{19}   - 2 
Hence the sum of the series  =2+2^{19}-2 = 2^{19}

Option  A

BrentGMATPrepNow · Answer

pushpitkc's solution is the same as mine (look for a pattern). 
However, 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 + 2^18 
Notice 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)  <  2^22

So, we can conclude that:  2 + 2^1 + 2^2 + 2^3 + . . . . 2^17 + 2^18  <  2^22

Only answer choice A is less than  2^22

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VodkaHelps · Answer

Wow, this question made it confusing because of the additional 2^1. At first I did not realize there was an extra 2 in power of 1. Therefore, I couldnt' find my answer. Later on realizing that I confidently picked A = 2^19. Use the formula of the geometric progression and you should be fine.

Gumbo_Stoic · Answer

2^n = 2 + 2^1 + ... + 2^n-1

Works with every other number as well.

Solution A

What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?

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What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?

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