# 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 $$n\in Odd\quad numbers$$

2 years, 8 months 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 $$n$$.

- 9 months, 3 weeks ago