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655-705 (Hard)|   Sequences|      
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gmihir
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A special sequence of numbers is written as 2, 9, 28, 65, 12 22 May 2012, 09:16
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

65% (hard)

Question Stats:

65% (02:41) correct 35% (02:44) wrong based on 312 sessions

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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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D
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A special sequence of numbers is written as 2, 9, 28, 65, 12 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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A special sequence of numbers is written as 2, 9, 28, 65, 12 24 May 2012, 10:17
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gmihir
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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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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A special sequence of numbers is written as 2, 9, 28, 65, 12 13 Jan 2014, 10:39
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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...
Show more

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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A special sequence of numbers is written as 2, 9, 28, 65, 12 15 Jan 2026, 02:38
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gmihir
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 :(
Show more
Deconstructing the Question
Sequence: 2, 9, 28, 65, 126...
Goal: Find the ratio of the 7th term to the 15th term.

Step 1: Identify the Pattern
Let's compare the term value to its position \(n\):
\(n=1 \rightarrow 2 = 1^3 + 1\)
\(n=2 \rightarrow 9 = 2^3 + 1\)
\(n=3 \rightarrow 28 = 3^3 + 1\)

General Formula: \(T_n = n^3 + 1\).

Step 2: Calculate the Specific Terms
We need the 7th term (\(T_7\)) and the 15th term (\(T_{15}\)).
Using the sum of cubes factorization \(a^3 + 1 = (a+1)(a^2 - a + 1)\) makes simplification easier.

7th Term:
\(T_7 = 7^3 + 1 = (7+1)(7^2 - 7 + 1) = 8(43)\).
(Alternatively: \(343 + 1 = 344\))

15th Term:
\(T_{15} = 15^3 + 1 = (15+1)(15^2 - 15 + 1) = 16(211)\).
(Alternatively: \(3375 + 1 = 3376\))

Step 3: Simplify the Ratio
\(\text{Ratio} = \frac{T_7}{T_{15}} = \frac{8 \cdot 43}{16 \cdot 211}\)

Cancel out 8 from top and bottom:
\(\text{Ratio} = \frac{1 \cdot 43}{2 \cdot 211} = \frac{43}{422}\)

Answer: D
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