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A sequence S is defined as follows: S_{n} = S_{n+1} + S_{n-1}/2. If S
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14 Feb 2021, 22:43
14 Feb 2021, 22:43
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A sequence S is defined as follows: \(S_{n} = \frac{S_{n+1} + S_{n-1}}{2}\). If \(S_{1} = 15\) and \(S_{4} = 10.5\), what is \(S_{2}\)?
A. 8
B. 10
C. 11.5
D. 12
E. 13.5
A. 8
B. 10
C. 11.5
D. 12
E. 13.5
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A sequence S is defined as follows: S_{n} = S_{n+1} + S_{n-1}/2. If S
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15 Feb 2021, 00:55
15 Feb 2021, 00:55
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GeminiHeat wrote:
A sequence S is defined as follows: \(S_{n} = \frac{S_{n+1} + S_{n-1}}{2}\). If \(S_{1} = 15\) and \(S_{4} = 10.5\), what is \(S_{2}\)?
A. 8
B. 10
C. 11.5
D. 12
E. 13.5
A. 8
B. 10
C. 11.5
D. 12
E. 13.5
There are 2 approaches for this question:
Approach1:
Given: \(S_{1} = 15\) and \(S_{4} = 10.5\)
\(S_{2} = \frac{S_{3} + S_{1}}{2}\) .......... (1)
\(S_{3} = \frac{S_{4} + S_{2}}{2}\)
\(2S_{3} = S_{4} + S_{2}\) ...........(2)
Plugging the value of \(S_{2}\) from (1) to (2);
\(2S_{3} = S_{4} + \frac{S_{3} + S_{1}}{2}\)
\(4S_{3} = 2S_{4} + S_{3} + S_{1}\)
\(3S_{3} = 2S_{4} + S_{1}\)
\(3S_{3} = 2(10.5) + 15\)
\(S_{3} = 36\)
\(S_{3} = 12\)
Now, \(S_{2} = \frac{S_{3} + S_{1}}{2}\)
\(S_{2} = \frac{12 + 15}{2} = 13.5\)
Approach2:
Any term in this sequence is the average of term after and before it, which means they are consecutive and the series must be an A.P
Series: \(15, S_{2}, S_{3}, 10.5\)
\(\frac{{15 + 10.5}}{2} = \frac{S_{2} + S_{3}}{2}\)
\(S_{2} + S_{3} = 25.5\) ....... (1)
Also, \(S_{2} - 15 = S_{3} - S_{2}\)
\(2S_{2} - S_{3} = 15\) ....... (2)
Solve (1) and (2);
\(S_{2} = 13.5\)
Hence, option E
Re: A sequence S is defined as follows: S_{n} = S_{n+1} + S_{n-1}/2. If S
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02 Jun 2025, 02:21
02 Jun 2025, 02:21
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