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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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13 Aug 2017, 00:12
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67% (01:02) correct 33% (01:03) wrong based on 57 sessions

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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) $$2 + 2^{18!}$$
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Re: What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18? [#permalink]

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13 Aug 2017, 00:29
1
$$2 + 2^1 = 2 + 2 = 4 = 2^2$$

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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13 Aug 2017, 08:06
Top Contributor
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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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
What is the value of 2 + 2^1 + 2^2 + 2^3...........2^18?   [#permalink] 13 Aug 2017, 10:51
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