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Sn in terms of n

Let: \({ S }_{ n }=2n-3+{ S }_{ n-2 }\) for \(n>1\)

Find a rule for \({ S }_{ n }\) in terms of \(n\) if \({ S }_{ 1 }=1\), where \(n\in Odd\quad numbers\)

Note by Vladimir Smith
2 years ago

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If \(S_n\) is a polynomial, the degree of \(S_n-S_{n-2}=2n-3\) is one less than the degree of \(S_n\), making \(S_n\) quadratic.

Since \(\dfrac{1}{2}n^2-\dfrac{1}{2}(n-2)^2-\bigg(\dfrac{1}{2}n-\dfrac{1}{2}(n-2)\bigg)=2n-3\), and \(S_1=1\), we have that \(S_n=\dfrac{1}{2}n^2-\dfrac{1}{2}n+1\) will work for all positive, odd \(n\).

Miles Koumouris - 1 month, 4 weeks ago

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