Question

An arithmetic sequence is defined by the general term tn = 19 - (n - 1)18, where n ∈N and n ≥ 1. What is the recursive formula of the sequence?

Answer 1

Answer:

$t_1= \text{19 for n =1}$

$t_n=t_{n-1}-18 \text{ for n }> 1$

Step-by-step explanation:

Given : General term of  arithmetic sequence : $t_n=19-18(n-1)$

To Find:  the recursive formula of the sequence

Solution :

$t_n=19-18(n-1)$

To find the recursive formula of the sequence .

$t_{n-1}=19-18(n-1-1)$

$t_{n-1}=19-18(n-2)$

So, the recursive formula is :

$t_n-t_{n-1}=19-18(n-1)-[19-18(n-2)]$

$t_n-t_{n-1}=19-18n+18-{19-18n+36]$

$t_n-t_{n-1}=19-18n+18-19+18n-36$

$t_n-t_{n-1}=-18$

$t_n=t_{n-1}-18$

Hence the recursive formula of the sequence is :

$t_1= \text{19 for n =1}$

$t_n=t_{n-1}-18 \text{ for n }> 1$

Answer 2

$t_n=19-18(n-1)$
$t_{n-1}=19-18(n-2)$
$\implies t_n-t_{n-1}=-18(n-1)+18(n-2)$
$\implies t_n=t_{n-1}-18n+18+18n-36$
$\implies t_n=t_{n-1}-18$

So the recursive formula is

$\begin{cases}t_1=19\\t_n=t_{n-1}-18&\text{for }n>1\end{cases}$