To determine a sequence defined by the recursive formula [tex]\( f(n+1) = f(n) + 2.5 \)[/tex] for [tex]\( n \geq 1 \)[/tex], we need to start with an initial term and then apply the given formula iteratively to generate subsequent terms.
Let's assume the initial term of the sequence is [tex]\( f(1) = 0 \)[/tex].
Using the recursive formula, we'll compute the next few terms step-by-step:
1. First term:
[tex]\[
f(1) = 0
\][/tex]
2. Second term:
[tex]\[
f(2) = f(1) + 2.5 = 0 + 2.5 = 2.5
\][/tex]
3. Third term:
[tex]\[
f(3) = f(2) + 2.5 = 2.5 + 2.5 = 5.0
\][/tex]
4. Fourth term:
[tex]\[
f(4) = f(3) + 2.5 = 5.0 + 2.5 = 7.5
\][/tex]
5. Fifth term:
[tex]\[
f(5) = f(4) + 2.5 = 7.5 + 2.5 = 10.0
\][/tex]
6. Sixth term:
[tex]\[
f(6) = f(5) + 2.5 = 10.0 + 2.5 = 12.5
\][/tex]
7. Seventh term:
[tex]\[
f(7) = f(6) + 2.5 = 12.5 + 2.5 = 15.0
\][/tex]
8. Eighth term:
[tex]\[
f(8) = f(7) + 2.5 = 15.0 + 2.5 = 17.5
\][/tex]
9. Ninth term:
[tex]\[
f(9) = f(8) + 2.5 = 17.5 + 2.5 = 20.0
\][/tex]
10. Tenth term:
[tex]\[
f(10) = f(9) + 2.5 = 20.0 + 2.5 = 22.5
\][/tex]
So, the first 10 terms of the sequence defined by the recursive formula [tex]\( f(n+1) = f(n) + 2.5 \)[/tex] starting with [tex]\( f(1) = 0 \)[/tex] are:
[tex]\[ 0, 2.5, 5.0, 7.5, 10.0, 12.5, 15.0, 17.5, 20.0, 22.5 \][/tex]
