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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) _________________
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Re: What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?
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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 nineteen2^18'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)
What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?
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13 Aug 2017, 10:51
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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) \(2 + 2^{18!}\)
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 \(=2+2^{19}-2 = 2^{19}\)
Re: What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?
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01 Oct 2018, 02:37
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.
gmatclubot
Re: What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?
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01 Oct 2018, 02:37