Skip to main content

Solve aₙ=2aₙ₋₁+3ⁿ

Short answer

Homogeneous part α·2^n. The term 3^n is not a root, so particular solution 3^{n+1}. With a_1=5, α=-2, resulting in a_n = 3^{n+1} - 2^{n+1}.

Find all solutions of the recurrence relation aₙ = 2aₙ₋₁ + 3ⁿ with initial condition a₁ = 5.

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

Step 1. Homogeneous part

r = 2, so aₙ⁽ʰ⁾ = α·2ⁿ.

Step 2. Particular solution

F(n) = 3ⁿ and 3 is not a root, so try aₙ⁽ᵖ⁾ = c·3ⁿ.

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

So aₙ⁽ᵖ⁾ = 3·3ⁿ = 3ⁿ⁺¹.

Step 3. All solutions

aₙ = α·2ⁿ + 3ⁿ⁺¹, α any constant.

Step 4. Initial condition a₁ = 5

2α + 9 = 5, so α = −2.

Answer: aₙ = 3ⁿ⁺¹ − 2ⁿ⁺¹

Check: a₁ = 9 − 4 = 5 ✓; a₂ = 27 − 8 = 19 = 2(5) + 9 ✓; a₃ = 81 − 16 = 65 = 2(19) + 27 ✓.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Discrete Structure chapter 3 questions →