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Use the recursive formula to find the first five terms in the arithmetic sequence.

[tex]\[
\begin{array}{l}
f(1)=54 \\
f(n)=f(n-1)-9
\end{array}
\][/tex]

A. [tex][tex]$54, 63, 72, 81, 90$[/tex][/tex]
B. [tex][tex]$45, 36, 27, 18, 9$[/tex][/tex]
C. [tex][tex]$45, 54, 63, 72, 81$[/tex][/tex]
D. [tex][tex]$54, 45, 36, 27, 18$[/tex][/tex]



Answer :

GinnyAnswer
To solve the problem, we need to generate the first five terms of the arithmetic sequence defined by the given recursive formula:

1. Starting Term (Initial Condition): The first term is provided as [tex]\( f(1) = 54 \)[/tex].

2. Recursive Formula: The formula that defines each term based on the previous term is [tex]\( f(n) = f(n-1) - 9 \)[/tex].

Let's determine each term step by step:

- Step 1: The first term [tex]\( f(1) \)[/tex] is given as 54.
[tex]\[ f(1) = 54 \][/tex]
The sequence now is: [tex]\( [54] \)[/tex].

- Step 2: To find the second term [tex]\( f(2) \)[/tex], we use the recursive formula with [tex]\( n = 2 \)[/tex]:
[tex]\[ f(2) = f(1) - 9 = 54 - 9 = 45 \][/tex]
The sequence now is: [tex]\( [54, 45] \)[/tex].

- Step 3: To find the third term [tex]\( f(3) \)[/tex], we use the recursive formula with [tex]\( n = 3 \)[/tex]:
[tex]\[ f(3) = f(2) - 9 = 45 - 9 = 36 \][/tex]
The sequence now is: [tex]\( [54, 45, 36] \)[/tex].

- Step 4: To find the fourth term [tex]\( f(4) \)[/tex], we use the recursive formula with [tex]\( n = 4 \)[/tex]:
[tex]\[ f(4) = f(3) - 9 = 36 - 9 = 27 \][/tex]
The sequence now is: [tex]\( [54, 45, 36, 27] \)[/tex].

- Step 5: To find the fifth term [tex]\( f(5) \)[/tex], we use the recursive formula with [tex]\( n = 5 \)[/tex]:
[tex]\[ f(5) = f(4) - 9 = 27 - 9 = 18 \][/tex]
The sequence now is: [tex]\( [54, 45, 36, 27, 18] \)[/tex].

Thus, the first five terms of the sequence are:
[tex]\[ 54, 45, 36, 27, 18 \][/tex]

Given the choices:
1. [tex]\( 54, 63, 72, 81, 90 \)[/tex]
2. [tex]\( 45, 36, 27, 18, 9 \)[/tex]
3. [tex]\( 45, 54, 63, 72, 81 \)[/tex]
4. [tex]\( 54, 45, 36, 27, 18 \)[/tex]

The correct answer is:
[tex]\[ 54, 45, 36, 27, 18 \][/tex]

So, the correct choice is the fourth option.

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