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Given:
[tex]\[ A(n+1) = A(n) - 19 \][/tex]
for \( n \geq 1 \) and \( A(1) = -6 \).

Find [tex]\( A(n) \)[/tex].

Sagot :

GinnyAnswer
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.
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