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# 2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) =

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2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) =  [#permalink]

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Updated on: 06 Nov 2017, 00:46
9
00:00

Difficulty:

55% (hard)

Question Stats:

61% (01:17) correct 39% (01:45) wrong based on 131 sessions

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$$2^{(22)} + 2^{(23)} + 2^{(24)} + ... + 2^{(43)} + 2^{(44)} =$$

A $$2^{22}(2^{(23)}-1)$$

B $$2^{22}(2^{(24)}-1)$$

C $$2^{22}(2^{(25)}-1)$$

D $$2^{23}(2^{(21)}-1)$$

E $$2^{23}(2^{(22)}-1)$$

Originally posted by arabella on 06 Nov 2017, 00:39.
Last edited by Bunuel on 06 Nov 2017, 00:46, edited 1 time in total.
Edited the question and added the OA.
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Joined: 02 Sep 2009
Posts: 52343
2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) =  [#permalink]

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06 Nov 2017, 01:02
3
2
arabella wrote:
$$2^{(22)} + 2^{(23)} + 2^{(24)} + ... + 2^{(43)} + 2^{(44)} =$$

A $$2^{22}(2^{(23)}-1)$$

B $$2^{22}(2^{(24)}-1)$$

C $$2^{22}(2^{(25)}-1)$$

D $$2^{23}(2^{(21)}-1)$$

E $$2^{23}(2^{(22)}-1)$$

You can solve this question by applying the GP formula.

$$2^{(22)} + 2^{(23)} + 2^{(24)} + ... + 2^{(43)} + 2^{(44)} =2^{22}(1+2+2^2+...+2^{22})=$$

Now, again you can apply the formula for GP or just observe the pattern:

$$1 + 2 = 3 =4-1 =2^2 - 1$$;
$$1 + 2 + 2^2 = 7 = 8-1=2^3 - 1$$;
$$1 + 2 + 2^2 + 2^3 = 15 = 16-1=2^4 - 1$$;
...
$$1+2+2^2+...+2^{22} = 2^{23} - 1$$.

Thus, $$2^{22}(1+2+2^2+...+2^{22})=2^{22}(2^{23} - 1)$$.

Similar question: https://gmatclub.com/forum/36-126078.html
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Re: 2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) =  [#permalink]

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06 Nov 2017, 01:15
arabella wrote:
$$2^{(22)} + 2^{(23)} + 2^{(24)} + ... + 2^{(43)} + 2^{(44)} =$$

A $$2^{22}(2^{(23)}-1)$$

B $$2^{22}(2^{(24)}-1)$$

C $$2^{22}(2^{(25)}-1)$$

D $$2^{23}(2^{(21)}-1)$$

E $$2^{23}(2^{(22)}-1)$$

Its a GP Series, a = 2^22, r = 2, n=23

S = a(r^n-1)/r-1
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Re: 2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) =  [#permalink]

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06 Nov 2017, 01:52
arabella wrote:
$$2^{(22)} + 2^{(23)} + 2^{(24)} + ... + 2^{(43)} + 2^{(44)} =$$

A $$2^{22}(2^{(23)}-1)$$

B $$2^{22}(2^{(24)}-1)$$

C $$2^{22}(2^{(25)}-1)$$

D $$2^{23}(2^{(21)}-1)$$

E $$2^{23}(2^{(22)}-1)$$

a different way and a point to remember.....
$$2^1 + 2^2 + 2^3 .........+ 2^{n-1} + 2^n=2^{n+1}-2$$

so...
$$2^{(22)} + 2^{(23)} + 2^{(24)} + ... + 2^{(43)} + 2^{(44)} = (2^1+2^2+......2^{44})-(2^1+2^2+....+2^{21}$$
as per above formula
$$(2^1+2^2+......2^{44})-(2^1+2^2+....+2^{21}=(2^{(44+1)}-2)-(2^{(21+1)}-2)=2^{(44+1)}-2-2^{(21+1)}+2=2^{(45)}-2^{(22)}=2^{22}(2^{23}-1)$$
A

you can check why $$2^1 + 2^2 + 2^3 .........+ 2^{n-1} + 2^n=2^{n+1}-2$$ at
https://gmatclub.com/forum/2-252919.html
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2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: 2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) =  [#permalink]

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08 Nov 2017, 16:26
arabella wrote:
$$2^{(22)} + 2^{(23)} + 2^{(24)} + ... + 2^{(43)} + 2^{(44)} =$$

A $$2^{22}(2^{(23)}-1)$$

B $$2^{22}(2^{(24)}-1)$$

C $$2^{22}(2^{(25)}-1)$$

D $$2^{23}(2^{(21)}-1)$$

E $$2^{23}(2^{(22)}-1)$$

Let’s begin by factoring out 2^22, which is common to each of the terms:

2^22 + 2^23 + … + 2^44 = 2^22(1 + 2 + 2^2 + … + 2^22)

To find an expression for 1 + 2 + 2^2 + … + 2^22, we will use the formula

a^n - 1 = (a - 1)(a^(n - 1) + a^(n - 2) + … + a + 1).

If we let a = 2 and n = 23, we obtain

2^23 - 1 = (2 - 1)(2^22 + 2^21 + … + 2 + 1).

Thus:

2^22 + 2^23 + … + 2^44 = 2^22(1 + 2 + 2^2 + … + 2^22) = 2^22(2^23 - 1)

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Re: 2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) =  [#permalink]

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23 Dec 2018, 08:35
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Re: 2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) = &nbs [#permalink] 23 Dec 2018, 08:35
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