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.
