If a, start subscript, 1, end subscript, equals, 5*a*1​=5,   a, start subscript, 2, end subscript, equals, 0*a*2​=0 and a, start subscript, n, end subscript, equals, 3, a, start subscript, n, minus, 1, end subscript, plus, a, start subscript, n, minus, 2, end subscript*an*​=3*an*−1​+*an*−2​ then find the value of a, start subscript, 5, end subscript*a*5​.

The given recurrence relation is:

\[
a_1 = 5,\quad a_2 = 0,\quad a_n = 3a_{n-1} + a_{n-2} \text{ for } n \geq 3
\]

We are tasked with finding \( a_5 \). Let’s calculate each term step by step:

1. We know \( a_1 = 5 \) and \( a_2 = 0 \).
2. Using the recurrence relation for \( a_3 \):
   \[
   a_3 = 3a_2 + a_1 = 3(0) + 5 = 5
   \]
3. Next, for \( a_4 \):
   \[
   a_4 = 3a_3 + a_2 = 3(5) + 0 = 15
   \]
4. Finally, for \( a_5 \):
   \[
   a_5 = 3a_4 + a_3 = 3(15) + 5 = 45 + 5 = 50
   \]

Thus, the value of \( a_5 \) is \( \boxed{50} \).

Would you like any further clarification on this, or more details? Here are some related questions:

1. How do you find terms of a sequence using recurrence relations?
2. What are the different types of recurrence relations?
3. How can recurrence relations be solved explicitly?
4. What are some common applications of recurrence relations in mathematics?
5. What is the characteristic equation of a linear recurrence relation?

**Tip**: When working with recurrence relations, it's useful to compute the first few terms manually to understand the pattern!

If a₁ = 5, a₂ = 0 and aₙ = 3aₙ₋₁ + aₙ₋₂, then find the value of a₅.

Learn how to find the value of a₅ in a sequence defined by the recurrence relation aₙ = 3aₙ₋₁ + aₙ₋₂, with initial terms a₁ = 5 and a₂ = 0. This problem explores key concepts in sequences and algebra, suitable for students in Grades 9-11.

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