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Answer :
Sure, let's find the function that determines the [tex]\( n \)[/tex]th term of the sequence [tex]\(52, 48, 44, 40, 36\)[/tex].
First, we observe that the sequence is decreasing by 4 each time. This indicates that the sequence is an arithmetic sequence where the common difference [tex]\( d \)[/tex] is [tex]\(-4\)[/tex].
The general formula for the [tex]\( n \)[/tex]th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where [tex]\( a_1 \)[/tex] is the first term, [tex]\( d \)[/tex] is the common difference, and [tex]\( n \)[/tex] is the term number.
For our sequence:
[tex]\[ a_1 = 52 \][/tex]
[tex]\[ d = -4 \][/tex]
Plugging these values into the general formula, we get:
[tex]\[
a_n = 52 + (n-1) \cdot (-4)
\][/tex]
Now, let's simplify this expression step-by-step:
[tex]\[
a_n = 52 + (n-1) \cdot (-4)
\][/tex]
[tex]\[
a_n = 52 + (-4n + 4)
\][/tex]
[tex]\[
a_n = 52 - 4n + 4
\][/tex]
[tex]\[
a_n = 56 - 4n
\][/tex]
So the function that determines the [tex]\( n \)[/tex]th term of the sequence is:
[tex]\[
f(n) = 56 - 4n
\][/tex]
Now let's match this with the given options:
A. [tex]\( f(n) = 4n + 52 \)[/tex]
B. [tex]\( f(n) = -4n + 56 \)[/tex]
C. [tex]\( f(n) = 4n + 56 \)[/tex]
D. [tex]\( f(n) = -4n + 52 \)[/tex]
We see that the function we found [tex]\( f(n) = 56 - 4n \)[/tex] matches option B [tex]\( f(n) = -4n + 56 \)[/tex].
Therefore, the correct function is:
[tex]\[
\boxed{B \; f(n) = -4n + 56}
\][/tex]
First, we observe that the sequence is decreasing by 4 each time. This indicates that the sequence is an arithmetic sequence where the common difference [tex]\( d \)[/tex] is [tex]\(-4\)[/tex].
The general formula for the [tex]\( n \)[/tex]th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where [tex]\( a_1 \)[/tex] is the first term, [tex]\( d \)[/tex] is the common difference, and [tex]\( n \)[/tex] is the term number.
For our sequence:
[tex]\[ a_1 = 52 \][/tex]
[tex]\[ d = -4 \][/tex]
Plugging these values into the general formula, we get:
[tex]\[
a_n = 52 + (n-1) \cdot (-4)
\][/tex]
Now, let's simplify this expression step-by-step:
[tex]\[
a_n = 52 + (n-1) \cdot (-4)
\][/tex]
[tex]\[
a_n = 52 + (-4n + 4)
\][/tex]
[tex]\[
a_n = 52 - 4n + 4
\][/tex]
[tex]\[
a_n = 56 - 4n
\][/tex]
So the function that determines the [tex]\( n \)[/tex]th term of the sequence is:
[tex]\[
f(n) = 56 - 4n
\][/tex]
Now let's match this with the given options:
A. [tex]\( f(n) = 4n + 52 \)[/tex]
B. [tex]\( f(n) = -4n + 56 \)[/tex]
C. [tex]\( f(n) = 4n + 56 \)[/tex]
D. [tex]\( f(n) = -4n + 52 \)[/tex]
We see that the function we found [tex]\( f(n) = 56 - 4n \)[/tex] matches option B [tex]\( f(n) = -4n + 56 \)[/tex].
Therefore, the correct function is:
[tex]\[
\boxed{B \; f(n) = -4n + 56}
\][/tex]
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