4+2^2+2^3+2^4+2^5+2^6+2^7=

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Fri Sep 29, 2017 6:37 pm

4+2^2+2^3+2^4+2^5+2^6+2^7=

A) 2^7
B) 2^8
C) 2^16
D) 2^28
E) 2^29

The OA is B.

Is there any way to make this PS question without using a calculator?

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Source: Beat The GMAT — Problem Solving | Fri Sep 29, 2017 6:37 pm

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by Brent@GMATPrepNow » Fri Sep 29, 2017 7:29 pm

Vincen wrote:4+2^2+2^3+2^4+2^5+2^6+2^7=

A) 2^7
B) 2^8
C) 2^16
D) 2^28
E) 2^29

Let's look for a pattern...

We want: 4 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 = ?

4 + 2^2 = 4 + 4 = 8 = 2^3
So, 4 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 = 2^3 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7

2^3 + 2^3 = 2(2^3) = 2^4
So, 2^3 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 = 2^4 + 2^4 + 2^5 + 2^6 + 2^7

2^4 + 2^4 = 2(2^4) = 2^5
So, 2^4 + 2^4 + 2^5 + 2^6 + 2^7 = 2^5 + 2^5 + 2^6 + 2^7

Continuing the pattern, we get: 4 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 = 2^8

Answer: B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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by [email protected] » Sat Sep 30, 2017 3:52 pm

Hi Vincen,

With a little math and a bit of logic, you can get to the correct answer here without too much trouble. To start, you have to recognize what it really means to raise 2 to higher and higher exponents. With each increase of 1 to the exponent, the total DOUBLES.

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
Etc.

Thus, at higher and higher exponents, the numbers are NOT close together. From the sum of terms given to us, we can see that the largest term will be 2^7 and (when we word 'down' the list) we're adding smaller and smaller numbers to it. Clearly the sum will be GREATER than 2^7, but not that much greater (relatively speaking). Looking at the answer choices, there's only one option that makes sense...

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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by EconomistGMATTutor » Tue Oct 10, 2017 1:20 pm

Vincen wrote:4+2^2+2^3+2^4+2^5+2^6+2^7=

A) 2^7
B) 2^8
C) 2^16
D) 2^28
E) 2^29

The OA is B.

Is there any way to make this PS question without using a calculator?

Hi Vincen,
Let's take a look at your question.

4+2^2+2^3+2^4+2^5+2^6+2^7
We can write the first term of the sum 4 as 2^2,
=(2^2+2^2)+2^3+2^4+2^5+2^6+2^7
= 2(2^2)+2^3+2^4+2^5+2^6+2^7

We know that 2(2^2) = 2^3, so
= (2^3+2^3)+2^4+2^5+2^6+2^7

Add 2^3 + 2^3 = 2(2^3)
= 2(2^3) + 2^4+2^5+2^6+2^7
= 2^4 + 2^4+2^5+2^6+2^7

Add 2^4 + 2^4 = 2(2^4)
= 2(2^4) +2^5+2^6+2^7
=( 2^5 + 2^5)+2^6+2^7

Add 2^5 + 2^5 = 2(2^5)
= 2(2^5) + 2^6+2^7
=( 2^6+2^6)+2^7

Add 2^6 + 2^6 = 2(2^6)
= 2(2^6)+2^7
= 2^7+2^7
= 2(2^7)
= 2^8

Therefore, Option B is correct.

I am available if you'd like any follow up.

GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.

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Re: 4+2^2+2^3+2^4+2^5+2^6+2^7=

by Scott@TargetTestPrep » Sun Feb 02, 2020 10:09 am

Vincen wrote: ↑
Fri Sep 29, 2017 6:37 pm
4+2^2+2^3+2^4+2^5+2^6+2^7=

A) 2^7
B) 2^8
C) 2^16
D) 2^28
E) 2^29

The OA is B.

Is there any way to make this PS question without using a calculator?

We can use the rule that 2^n + 2^n = 2^(n+1).

Starting with the first two terms, we have:

2^2 + 2^2 = 2^3

Then we have:

2^3 + 2^3 = 2^4

If we continue this pattern, we end with:

2^7 + 2^7 = 2^8.

Answer: B

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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