Find the solution to the following lhcc recurrence:
an=3nan−1 for n2 with initial conditions a0=4.
An = 3 n An-1 ??
A0 = 4 given
A1 = 3*4 = 12
A2 = 3*12 = 36
To find the solution to the given linear homogeneous constant coefficient recurrence (LHCC recurrence), we can use the following steps:
Step 1: Understand the recurrence relation.
The given recurrence relation is an = 3 * an-1 for n ≥ 2, with the initial condition a₀ = 4. This means that each term in the sequence is three times the previous term.
Step 2: Write out the terms of the sequence.
Based on the given recurrence relation and the initial condition, we can write out the first few terms of the sequence:
a₀ = 4
a₁ = 3 * a₀ = 3 * 4 = 12
a₂ = 3 * a₁ = 3 * 12 = 36
a₃ = 3 * a₂ = 3 * 36 = 108
...
Step 3: Observe the pattern.
By looking at the terms of the sequence, we can observe a pattern. Each term is obtained by multiplying the previous term by 3. This suggests that the general formula for an might involve powers of 3.
Step 4: Guess the general form of the solution.
Based on the pattern observed in Step 3, we can guess that the general formula for an is an = 3ⁿ * c, where c is a constant to be determined.
Step 5: Use the initial condition to find the constant.
To determine the value of the constant c, we substitute the initial condition a₀ = 4 into the general formula:
a₀ = 3⁰ * c = 1 * c = 4
This implies that c = 4.
Step 6: Determine the solution.
Using the value of c obtained in Step 5, we can now write the solution to the given recurrence:
an = 3ⁿ * c = 3ⁿ * 4, for n ≥ 0.
So, the solution to the given LHCC recurrence an = 3 * an-1 for n ≥ 2, with the initial condition a₀ = 4, is given by the formula:
an = 3ⁿ * 4, for n ≥ 0.