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A special sequence of numbers is written as 2, 9, 28, 65, 12

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gmihir
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A special sequence of numbers is written as 2, 9, 28, 65, 12 [#permalink] New post   22 May 2012, 09:16
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A special sequence of numbers is written as 2, 9, 28, 65, 126 …….Find the ratio of the 7th term to the 15th term.

A. 1/4
B. 1/2
C. 129/422
D. 43/422
E. 43/129

couldn't get the logic behind the terms in the sequence :(
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Cares
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Re: special sequence of numbers is written as 2, 9, 28, 65 [#permalink] New post   22 May 2012, 09:24
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Sequence is

\(1^3+1...2^3+1...3^3+1.....7^3+1.....15^3+1\)

Ratio thus is

\(\frac{7^3+1}{15^3+1}\)

Not sure what the quick way to simplify this is...
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gmatDeep
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Re: A special sequence of numbers is written as 2, 9, 28, 65, 12 [#permalink] New post   24 May 2012, 10:17
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gmihir wrote:
A special sequence of numbers is written as 2, 9, 28, 65, 126 …….Find the ratio of the 7th term to the 15th term.

A. 1/4
B. 1/2
C. 129/422
D. 43/422
E. 43/129

couldn't get the logic behind the terms in the sequence :(


The series as pointed out already is \((1^3+1) , (2^3+1) , (3^3+1) , (4^3+1)..\)
So, ratio of the 7th term to the 15th term = \(7^3+1/15^3+1\)

Simplify it as \((7+1)(7^2 - 7*1 + 1)/(15+1)(15^2 - 15*1 + 1)\)
because \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\)

So, our ratio is \(8*43/16*211 = 43/422\)

Answer D.
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Re: special sequence of numbers is written as 2, 9, 28, 65 [#permalink] New post   13 Jan 2014, 10:39
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Cares wrote:
Sequence is

\(1^3+1...2^3+1...3^3+1.....7^3+1.....15^3+1\)

Ratio thus is

\(\frac{7^3+1}{15^3+1}\)

Not sure what the quick way to simplify this is...


Well I'll tel you what I did

You end up with 344/3375

I just used what I call 'Brute Force Heavy discount shorcut' and ended up with 3/33 = 1/11 approx

So among the answer choices the only one that made sence was D

Not sure if there is another approach that might be more elegant

Cheers!
J :)

PS. After reflecting on this...

Actually what one can do is the following:

Once you get to (7/15)^3 then this is approx 1/8

Only answer choice that even comes close to 1/8 is answer choice D

Hence D it is!
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Re: A special sequence of numbers is written as 2, 9, 28, 65, 12 [#permalink] New post   31 Jan 2024, 11:19
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Re: A special sequence of numbers is written as 2, 9, 28, 65, 12 [#permalink]
31 Jan 2024, 11:19
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