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Solve aₙ=3aₙ₋₁+2ⁿ

Short answer

Homogeneous part α·3^n. The nonhomogeneous term 2^n is not a root, so particular solution -2^{n+1}. Using a_0=5 gives α=7, yielding a_n = 7·3^n - 2^{n+1}.

Find all solutions of the recurrence relation aₙ = 3aₙ₋₁ + 2ⁿ with initial condition a₀ = 5.

aₙ = 3aₙ₋₁ + 2ⁿ is linear, non-homogeneous, of order 1.

Step 1. Homogeneous part

aₙ = 3aₙ₋₁ gives r = 3, so aₙ⁽ʰ⁾ = α·3ⁿ.

Step 2. Particular solution

F(n) = 2ⁿ and 2 is not a root, so try aₙ⁽ᵖ⁾ = c·2ⁿ.

c·2ⁿ = 3c·2ⁿ⁻¹ + 2ⁿ      (divide by 2ⁿ⁻¹)
2c   = 3c + 2
c    = -2

So aₙ⁽ᵖ⁾ = −2·2ⁿ = −2ⁿ⁺¹.

Step 3. All solutions

aₙ = α·3ⁿ − 2ⁿ⁺¹, α any constant.

Step 4. Initial condition a₀ = 5

α − 2 = 5, so α = 7.

Answer: aₙ = 7·3ⁿ − 2ⁿ⁺¹

Check: a₁ = 21 − 4 = 17 = 3(5) + 2 ✓; a₂ = 63 − 8 = 55 = 3(17) + 4 ✓; a₃ = 189 − 16 = 173 = 3(55) + 8 ✓.

✓ Answer checked by Claude Opus 5.5 (high effort)

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