The terms of a sequence are defined by `a_(n)=3a_(n-1)-a_(n-2)` for `n gt 2`. What is the value of `a_(5)` if `a_(1)=4` and `a_(2)=3` ?

Options

A. 12
B. 23
C. 25
D. 31

Explanation

To find the value of \( a_5 \) in the sequence defined by the recurrence relation \( a_n = 3a_{n-1} - a_{n-2} \) for \( n > 2 \), with the initial conditions \( a_1 = 4 \) and \( a_2 = 3 \), we will calculate the terms step by step.

### Step 1: Calculate \( a_3 \)
Using the recurrence relation for \( n = 3 \):
\[ a_3 = 3a_{2} - a_{1} \]
Substituting the known values:
\[ a_3 = 3(3) - 4 = 9 - 4 = 5 \]

### Step 2: Calculate \( a_4 \)
Now, using the recurrence relation for \( n = 4 \):
\[ a_4 = 3a_{3} - a_{2} \]
Substituting the known values:
\[ a_4 = 3(5) - 3 = 15 - 3 = 12 \]

### Step 3: Calculate \( a_5 \)
Finally, using the recurrence relation for \( n = 5 \):
\[ a_5 = 3a_{4} - a_{3} \]
Substituting the known values:
\[ a_5 = 3(12) - 5 = 36 - 5 = 31 \]

Thus, the value of \( a_5 \) is \( \boxed{31} \).

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