Derivatives
\[ \begin{array} {c|c|c|c|c|c|c|c} n & 1 & 2 & 3 & 4 & 5 & 6 & \cdots \\ a_n = n^2 & \color{red}{1} & \color{red}{4} & \color{red}{9} & 16 & 25 & 36 & \cdots \\ \Delta(a_n) = a_{n+1} - a_n & \color{blue}{3} & \color{blue}{5} & ???\\ \end{array} \]
The second row of the table above shows the first few terms of the sequence \(a_n = n^2.\) Each entry in the third row is the difference between the next term in row 2 and the current term in row 2. For example, \(\color{blue}{3} = \color{red}{4} -\color{red}{1}\) and \(\color{blue}{5} = \color{red}{9} - \color{red}{4}.\) What number should go in the cell marked '???'?
\[ \begin{array} {c|c|c|c|c|c|c|c} n & 1 & 2 & 3 & 4 & 5 & 6 & \cdots \\ a_n = n^3 & 1 & 8 & 27 & 64 & 125 & 216 & \cdots \\ \Delta(a_n) = a_{n+1} - a_n & 7 & 19 & 37 & 61 & ???\\ \end{array} \]
The second row of the table above shows the first few terms of the sequence \(a_n = n^3.\) Each entry in the third row is the difference between the next term in row 2 and the current term in row 2. For example, \(7 = 8 -1\) and \(19 = 27 -8.\) What number should go in the cell marked '???'?
\[ \begin{array} {c|c|c|c|c|c|c|c} n & 1 & 2 & 3 & 4 & 5 & 6 & \cdots & n & n +1 \\ a_n = n^3 & 1 & 8 & 27 & 64 & 125 & 216 & \cdots &n^3 & (n+1)^3 \\ \Delta(a_n) = a_{n+1} - a_n & 7 & 19 & 37 & 61 & 91 & & \cdots & ? \\ \end{array} \]
Given \(a_n = n^3,\) find \(\Delta(a_n)\).
In other words, what is the general formula for the discrete derivative of \(\{n^3\}?\)
Given \(a_n = n^2 + n,\) find \(\Delta(a_n),\) the discrete derivative of \(\{a_n\}.\)
Which of these sequences has a constant discrete derivative?