Write recursive rule for sequence
1)
7, 20, 33, 46, 59.....
\(\begin{array}{|rcll|} \hline & \text{difference} \\ \hline 7 \\ & \color{red}13 \\ 20 \\ & \color{red}13 \\ 33 \\ & \color{red}13 \\ 46 \\ & \color{red}13 \\ 59 \\ \hline \end{array}\)
This is a Arithmetic Sequence
\(\text{Formula:}~ \boxed{a_n=a_1 + (n-1)d,\text{ with } a_1=7\text{ and }d = 13 }\)
\(\begin{array}{ll} \text{where:}\\ & a_1~ \text{ is the first term, and} \\ & d~ \text{ is the difference between the terms (called the "common difference")} \\ \end{array}\)
Write recursive rule
\(\begin{array}{|rcll|} \hline a_n &=& a_1 + (n-1)d \\ a_{n+1} &=& a_1 + \Big((n+1)-1 \Big)d \\ \hline a_{n+1}-a_n &=& a_1 + \Big((n+1)-1 \Big)d - \Big( a_1 + (n-1)d \Big) \\ a_{n+1}-a_n &=& a_1 + nd - a_1 - (n-1)d \\ a_{n+1}-a_n &=& a_1 + nd - a_1 - nd +d \\ a_{n+1}-a_n &=& d \quad & | \quad +a_n \\ \mathbf{a_{n+1}} & \mathbf{=} & \mathbf{a_n+ d} \qquad d=\color{red}13 \\ \mathbf{a_{n+1}} & \mathbf{=} & \mathbf{a_n+\color{red}13} \\ \hline \end{array} \)
