Math Problem Solver

$$1,2,9,28,65,126,...$$The next term in this cubic sequence is $a_{7}=217$.

The nth term in this sequence is given by the formula:

Explicit Formula: $$a_n=n \left(n^{2} - 3 n + 3\right)$$

Recursive Formula: $$a_n=a_{n-1} + 3 n^{2} - 9 n + 7, \text{where } a_{1}=1$$

$$\begin{matrix}1&&2&&9&&28&&65&&126\\[6pt]&+1&&+7&&+19&&+37&&+61\\[6pt]&&+6&&+12&&+18&&+24\\[6pt]&&&+6&&+6&&+6\\[6pt]\end{matrix}$$

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 $(x_{0}, x_{1}, x_{2}, x_{3})$: $$\begin{aligned} 1 &= 1^{3} x_{3} + 1^{2} x_{2} + 1 x_{1} + x_{0} \\ 2 &= 2^{3} x_{3} + 2^{2} x_{2} + 2 x_{1} + x_{0} \\ 9 &= 3^{3} x_{3} + 3^{2} x_{2} + 3 x_{1} + x_{0} \\ 28 &= 4^{3} x_{3} + 4^{2} x_{2} + 4 x_{1} + x_{0} \end{aligned} \quad \Rightarrow \quad \begin{aligned} x_{0} + x_{1} + x_{2} + x_{3} = 1\\x_{0} + 2 x_{1} + 4 x_{2} + 8 x_{3} = 2\\x_{0} + 3 x_{1} + 9 x_{2} + 27 x_{3} = 9\\x_{0} + 4 x_{1} + 16 x_{2} + 64 x_{3} = 28 \end{aligned}$$ $$ \Rightarrow \quad (x_{0}, x_{1}, x_{2}, x_{3})=\left( 0, \ 3, \ -3, \ 1\right) $$

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

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 $(x_{0}, x_{1}, x_{2})$: $$\begin{aligned} 2 &= 1 + 2^{2} x_{2} + 2 x_{1} + x_{0} \\ 9 &= 2 + 3^{2} x_{2} + 3 x_{1} + x_{0} \\ 28 &= 9 + 4^{2} x_{2} + 4 x_{1} + x_{0} \end{aligned} \quad \Rightarrow \quad \begin{aligned} x_{0} + 2 x_{1} + 4 x_{2} = 1\\x_{0} + 3 x_{1} + 9 x_{2} = 7\\x_{0} + 4 x_{1} + 16 x_{2} = 19 \end{aligned}$$ $$ \Rightarrow \quad (x_{0}, x_{1}, x_{2})=\left( 7, \ -9, \ 3\right) $$

The n^th term rule is:$$\begin{aligned} a_n&=a_{n-1} + n^{2} x_{2} + n x_{1} + x_{0} \\ &=a_{n-1} + n^{2} \left(3\right) + n \left(-9\right) + \left(7\right) \\ &=a_{n-1} + 3 n^{2} - 9 n + 7 \end{aligned}$$