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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? [#permalink]

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New post 13 Aug 2017, 00:12
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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) \(2 + 2^{18!}\)
[Reveal] Spoiler: OA

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Re: What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18? [#permalink]

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\(2 + 2^1 = 2 + 2 = 4 = 2^2\)

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

Therefore, \(2 + 2^1 + 2^2 + 2^3 + ...... + 2^{17} = 2^{18}\)

The value of the expression \(2 + 2^1 + 2^2 + 2^3........... + 2^{17} + 2^{18}\) = \(2^{18} + 2^{18}\) = \(2^{18}(1+1)\) = \(2^{18} * 2\) =\(2^{19}\)(Option A)
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Re: What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18? [#permalink]

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New post 13 Aug 2017, 08:06
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aashishagarwal2 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) \(2 + 2^{18!}\)


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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What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18? [#permalink]

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New post 13 Aug 2017, 10:51
aashishagarwal2 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) \(2 + 2^{18!}\)


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

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What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?   [#permalink] 13 Aug 2017, 10:51
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