High School

This shows a sequence:

[tex]\[ 52, 48, 44, 40, 36 \][/tex]

Which function can be used to determine the [tex] n [/tex]th term of the sequence?

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]

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]