Let's solve the recurrence relation \( A(n+1) = A(n) - 19 \) for \( n \geq 1 \) with the initial condition \( A(1) = -6 \).
1. Initial Condition:
Start with the initial condition \( A(1) = -6 \). This gives us the first term:
[tex]\[
A(1) = -6
\][/tex]
2. Applying the Recurrence Relation:
To find \( A(2) \), use the recurrence relation \( A(n+1) = A(n) - 19 \) with \( n = 1 \):
[tex]\[
A(2) = A(1) - 19 = -6 - 19 = -25
\][/tex]
Next, to find \( A(3) \), use the recurrence relation with \( n = 2 \):
[tex]\[
A(3) = A(2) - 19 = -25 - 19 = -44
\][/tex]
Then, to find \( A(4) \), use the recurrence relation with \( n = 3 \):
[tex]\[
A(4) = A(3) - 19 = -44 - 19 = -63
\][/tex]
Finally, to find \( A(5) \), use the recurrence relation with \( n = 4 \):
[tex]\[
A(5) = A(4) - 19 = -63 - 19 = -82
\][/tex]
3. Summary of Results:
We have calculated the following terms:
[tex]\[
A(1) = -6, \quad A(2) = -25, \quad A(3) = -44, \quad A(4) = -63, \quad A(5) = -82
\][/tex]
Therefore, the sequence \( \{A(n)\} \) for \( n = 1, 2, 3, 4, 5 \) is:
[tex]\[
[-6, -25, -44, -63, -82]
\][/tex]
This sequence was generated by repeatedly applying the recurrence relation starting from the initial condition.
