smanthasmith
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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?

Respuesta :

LammettHash
[tex]t_n=19-18(n-1)[/tex]
[tex]t_{n-1}=19-18(n-2)[/tex]
[tex]\implies t_n-t_{n-1}=-18(n-1)+18(n-2)[/tex]
[tex]\implies t_n=t_{n-1}-18n+18+18n-36[/tex]
[tex]\implies t_n=t_{n-1}-18[/tex]

So the recursive formula is

[tex]\begin{cases}t_1=19\\t_n=t_{n-1}-18&\text{for }n>1\end{cases}[/tex]

Answer:

[tex]t_1= \text{19 for n =1}[/tex]

[tex]t_n=t_{n-1}-18 \text{ for n }> 1[/tex]

Step-by-step explanation:

Given : General term of  arithmetic sequence : [tex]t_n=19-18(n-1)[/tex]

To Find:  the recursive formula of the sequence

Solution :

[tex]t_n=19-18(n-1)[/tex]

To find the recursive formula of the sequence .

[tex]t_{n-1}=19-18(n-1-1)[/tex]

[tex]t_{n-1}=19-18(n-2)[/tex]

So, the recursive formula is :

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

[tex]t_n-t_{n-1}=19-18n+18-{19-18n+36][/tex]

[tex]t_n-t_{n-1}=19-18n+18-19+18n-36[/tex]

[tex]t_n-t_{n-1}=-18[/tex]

[tex]t_n=t_{n-1}-18[/tex]

Hence the recursive formula of the sequence is :

[tex]t_1= \text{19 for n =1}[/tex]

[tex]t_n=t_{n-1}-18 \text{ for n }> 1[/tex]

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