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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