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