Certainly! Let's solve this problem step-by-step together.
The problem asks us to combine the two functions, [tex]\( f(n) \)[/tex] and [tex]\( g(n) \)[/tex], to create an arithmetic sequence [tex]\( a(n) \)[/tex]. Then, we need to find the 12th term of this sequence.
1. Start with the given functions:
- [tex]\( f(n) = 25 \)[/tex]
- [tex]\( g(n) = 3(n - 1) \)[/tex]
2. Combine these functions to form the sequence [tex]\( a(n) \)[/tex]. The sequence formula is given by:
[tex]\[
a(n) = f(n) - g(n)
\][/tex]
3. Substitute the expressions for [tex]\( f(n) \)[/tex] and [tex]\( g(n) \)[/tex] into the formula:
[tex]\[
a(n) = 25 - 3(n - 1)
\][/tex]
4. Simplify the expression inside the sequence formula.
[tex]\[
a(n) = 25 - 3(n - 1)
\][/tex]
Distribute the 3 inside the parentheses:
[tex]\[
a(n) = 25 - 3n + 3
\][/tex]
Combine like terms:
[tex]\[
a(n) = 28 - 3n
\][/tex]
5. Now, we need to find the 12th term, denoted as [tex]\( a(12) \)[/tex].
6. Substitute [tex]\( n = 12 \)[/tex] into the formula for the sequence [tex]\( a(n) \)[/tex]:
[tex]\[
a(12) = 28 - 3(12)
\][/tex]
7. Calculate the value:
[tex]\[
a(12) = 28 - 36
\][/tex]
[tex]\[
a(12) = -8
\][/tex]
So, the 12th term of the sequence is [tex]\( -8 \)[/tex].
