Question

Select ALL the correct answers.Consider the arithmetic sequence below.-5, 4, 13, 22, 31, ...Select all functions that define the given sequence.Of(1) = -5, f(n) = -f(n - 1) + 9, for n 2 2f(1) = -5, f(n) = f(n - 1) + 9, for n z 2Of(1) = -5, f(n) = f(n-1) + 7, for n 2 2fin) = -5n + 9O fin) = 9n - 5fin) = 9n - 14

Answer 1

The rules that define the sequence are f(n) = 9n - 14 and f(1) = -5; f(n) = f(n - 1) + 9

How to determine the rules that define the sequence

From the question, we have the following parameters that can be used in our computation:

-5, 4, 13, 22, 31, ...

In the above sequence, we can see that 9 is added to the previous term to get the new term

This means that

First term, a = -5

Common difference, d = 9

The nth term is then represented as

f(n) = a + (n - 1) * d

Substitute the known values in the above equation, so, we have the following representation

f(n) = -5 + 9(n - 1)

f(n) = 9n - 14

The recursive sequence of the above is

f(1) = -5; f(n) = f(n - 1) + 9

Answer 2

From the given sequence ,we can note that the common difference between consecutive numbers is 9. Thats because

`\begin{gathered} 4-(-5)=9 \\ 13-4=9 \\ 22-13=9 \\ 31-22=9 \end{gathered}`

This means that

`f(n)-f(n-1)=9`

or equivalently,

`f(n)=f(n-1)+9\text{ for n}\ge2`

We can also obtain the same result by taking into account the common difference and the fact that the first number in the sequence is -5. So we can write

`f(n)=9n-5`

For instance,

`\begin{gathered} n=0\Rightarrow f(0)=-5 \\ n=1\Rightarrow f(1)=9-5=4 \\ n=2\Rightarrow f(2)=18-5=13 \\ n=3=f(3)=27-5=22 \\ n=4\Rightarrow f(4)=36-5=31 \end{gathered}`

Therefore, the functions that define the given sequence are:

`\begin{gathered} f(1)=-5,\text{ f\lparen n\rparen=f\lparen n-1\rparen+9 for n}\ge2 \\ and \\ f(n)=9n-5 \end{gathered}`

which correspond to options 2 and 5 from top to bottom.