Math Problem Solver

$$0,-7,-31,-81,-166,...$$This infinite sequence is a cubic sequence. The n^th term rule for the sequence is $a_n=- \frac{3 n^{3}}{2} + \frac{n^{2}}{2} + 2 n - 1$, where $n>0$. The recursive formula is $a_n=a_{n-1} - \frac{9 n^{2}}{2} + \frac{11 n}{2}$, where $n>1$ and $a_{1}=0$.

$$\begin{matrix}0&&-7&&-31&&-81&&-166\\[6pt]&-7&&-24&&-50&&-85\\[6pt]&&-17&&-26&&-35\\[6pt]&&&-9&&-9\\[6pt]\end{matrix}$$

Explicit Formula

Since there are 3 rows of differences, the formula for the sequence can be written as a polynomial with degree 3, where $n$ is the term number and $(x_{0}, x_{1}, x_{2}, x_{3})$ are the coefficients: $$a_n=n^{3} x_{3} + n^{2} x_{2} + n x_{1} + x_{0}$$

Using the first 4 terms in the sequence, create and solve the system of equations for $(x_{0}, x_{1}, x_{2}, x_{3})$: $$\begin{aligned} 0 &= 1^{3} x_{3} + 1^{2} x_{2} + 1 x_{1} + x_{0} \\ -7 &= 2^{3} x_{3} + 2^{2} x_{2} + 2 x_{1} + x_{0} \\ -31 &= 3^{3} x_{3} + 3^{2} x_{2} + 3 x_{1} + x_{0} \\ -81 &= 4^{3} x_{3} + 4^{2} x_{2} + 4 x_{1} + x_{0} \end{aligned} \quad \Rightarrow \quad \begin{aligned} x_{0} + x_{1} + x_{2} + x_{3} = 0\\x_{0} + 2 x_{1} + 4 x_{2} + 8 x_{3} = -7\\x_{0} + 3 x_{1} + 9 x_{2} + 27 x_{3} = -31\\x_{0} + 4 x_{1} + 16 x_{2} + 64 x_{3} = -81 \end{aligned}$$ $$ \Rightarrow \quad (x_{0}, x_{1}, x_{2}, x_{3})=\left( -1, \ 2, \ \frac{1}{2}, \ - \frac{3}{2}\right) $$

The n^th term rule is:$$\begin{aligned} a_n&=n^{3} x_{3} + n^{2} x_{2} + n x_{1} + x_{0} \\ &=n^{3} \left(- \frac{3}{2}\right) + n^{2} \left(\frac{1}{2}\right) + n \left(2\right) + \left(-1\right) \\ &=- \frac{3 n^{3}}{2} + \frac{n^{2}}{2} + 2 n - 1 \end{aligned}$$

Recursive Formula

Since there are 3 rows of differences, the formula for the sequence can be written as the sum of $a_{n-1}$ and polynomial with degree 2, where $n$ is the term number and $(x_{0}, x_{1}, x_{2})$ are the coefficients: $$a_n=a_{n-1} + n^{2} x_{2} + n x_{1} + x_{0}$$

Using the first 4 terms in the sequence, create and solve the system of equations for $(x_{0}, x_{1}, x_{2})$: $$\begin{aligned} -7 &= 0 + 2^{2} x_{2} + 2 x_{1} + x_{0} \\ -31 &= -7 + 3^{2} x_{2} + 3 x_{1} + x_{0} \\ -81 &= -31 + 4^{2} x_{2} + 4 x_{1} + x_{0} \end{aligned} \quad \Rightarrow \quad \begin{aligned} x_{0} + 2 x_{1} + 4 x_{2} = -7\\x_{0} + 3 x_{1} + 9 x_{2} = -24\\x_{0} + 4 x_{1} + 16 x_{2} = -50 \end{aligned}$$ $$ \Rightarrow \quad (x_{0}, x_{1}, x_{2})=\left( 0, \ \frac{11}{2}, \ - \frac{9}{2}\right) $$

The n^th term rule is:$$\begin{aligned} a_n&=a_{n-1} + n^{2} x_{2} + n x_{1} + x_{0} \\ &=a_{n-1} + n^{2} \left(- \frac{9}{2}\right) + n \left(\frac{11}{2}\right) + \left(0\right) \\ &=a_{n-1} - \frac{9 n^{2}}{2} + \frac{11 n}{2} \end{aligned}$$