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Each number SN in a sequence can be expressed as a function

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Each number SN in a sequence can be expressed as a function [#permalink] New post 15 Jun 2017, 07:33
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Each number \(S_N\) in a sequence can be expressed as a function of the preceding number (\(S_{N–1}\)) as follows: \(S_N\)= \(\frac{2}{3}\) \(S_{N–1}\) \(- 4\). Which of the following equations correctly expresses the value of SN in this sequence in terms of \(S_{N+2}\) ?

A) \(S_N\) = \(\frac{9}{4}\) \(S_{N+2}\) \(+18\)

B) \(S_N\) = \(\frac{4}{9}\)\(S_{N+2}\) \(+15\)

C) \(S_N\) = \(\frac{9}{4}\) \(S_{N+2}\) \(+ 15\)

D) \(S_N\) = \(\frac{4}{9}\) \(S_{N+2}\) \(- 8\)

E) \(S_N\) = \(\frac{2}{3}\) \(S_{N+2}\) \(-8\)
[Reveal] Spoiler: OA

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Last edited by Carcass on 27 Dec 2018, 03:20, edited 3 times in total.
Edited by Carcass
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Re: Each number SN in a sequence can be expressed as a function [#permalink] New post 20 Sep 2017, 05:39
Given the rule for \(S_n\), \(S_{n+2}=\frac{2}{3}S_{n+1}-4\). Then, using the same rule, we know that \(S_{n+1}=\frac{2}{3}S_n-4\) and we can substitute this in the expression for \(S_{n+2}\), which gives us \(S_{n+2}=\frac{2}{3}(\frac{2}{3}S_n-4)-4\). Using easy algebra, we get \(S_n=\frac{9}{4}S_{n+2}+15\). Answer C!
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Re: Each number SN in a sequence can be expressed as a function [#permalink] New post 27 Dec 2017, 03:36
Can I get a further explanation on this problem? I'm not understanding exactly how to solve.

Thank you,
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Re: Each number SN in a sequence can be expressed as a function [#permalink] New post 11 Dec 2018, 18:15
Basically the later number = consecutive earlier number in the sequence times 2/3 then subtracts by 4
Sn
Sn+1 = 2/3 Sn-4
Sn+2=2/3Sn+1−4 = 2/3(2/3Sn -4)-4 = 4/9Sn - 20/3
===> Sn = (Sn+2 + 20/3)*9/4
Sn = 9/4Sn+2 + 15
C is the answer
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Re: Each number SN in a sequence can be expressed as a function [#permalink] New post 11 Dec 2018, 23:27
Expert's post
Carcass wrote:




Each number \(S_N\) in a sequence can be expressed as a function of the preceding number (\(S_{N–1}\)) as follows: \(S_N\)= \(\frac{2}{3}\) \(S_{N–1}\) – 4. Which of the following equations correctly expresses the value of SN in this sequence in terms of SN+2?

A) \(S_N\) = \(\frac{9}{4}\) \(S_{N+2}\) +18

B) \(S_N\) = \(\frac{4}{9}\)\(S_{N+2}\) +15

C) \(S_N\) = \(\frac{9}{4}\) \(S_{N+2}\) + 15

D) \(S_N\) = \(\frac{4}{9}\) \(S_{N+2}\) - 8

E) \(S_N\) = \(\frac{2}{3}\) \(S_{N+2}\) -8


let u swrite the \(S_N\)= \(\frac{2}{3}\) \(S_{N–1}\) – 4 in terms of N+2....
\(S_{N+2}\)= \(\frac{2}{3}\) \(S_{N+1}\) – 4, but \(S_{N+1}\)= \(\frac{2}{3}\) \(S_{N}\) – 4, so substitute this value in the previous equation..

\(S_{N+2}\)= \(\frac{2}{3}\) (\(\frac{2}{3}\) \(S_{N}\) – 4) – 4 =>\(S_{N+2}\)= \(\frac{2*2}{3*3}\) \(S_{N}-\frac{2*4}{3}\) – 4..
=> \(S_{N+2}\)= \(\frac{4}{9}\) \(S_{N+1}-\frac{8}{3}\) – 4,
Multiply the equation by 9..
\(9S_{N+2}\)= 4 \(S_{N}\)-8*3 –9* 4 => 4 \(S_{N}=9S_{N+2}\)+60
Divide the entire equation by 4 to get value of \(S_N\)
\(S_{N}=\frac{9}{4}S_{N+2}\)+15

C
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Re: Each number SN in a sequence can be expressed as a function   [#permalink] 11 Dec 2018, 23:27
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