---
title: "Solve aₙ=3aₙ₋₁+2ⁿ"
subject: "Discrete Structure"
chapter: 3
key: "semester_4/discrete_structure:recurrence-a-n-3a-2n"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/recurrence-a-n-3a-2n
site: BCT Engineering
---

# Solve aₙ=3aₙ₋₁+2ⁿ

Discrete Structure · Chapter 3: Recurrence Relation

- Asked in: 2071 Bhadra, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/recurrence-a-n-3a-2n
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-3

## Question

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

## 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}.

## Answer

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.
```text
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 ✓.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the homogeneous solution?
   A. α·2^n
   B. α·3^n
   C. α·n·3^n
   D. α + 3^n
   Answer: B. r = 3. (Key point: Homogeneous solution)

2. Why is c·2^n used?
   A. 2 is a double root
   B. 3 is a root
   C. 2^n is constant
   D. 2 is not a characteristic root
   Answer: D. No duplication, so no extra factor n. (Key point: Trial form)

3. What is c?
   A. −2
   B. 2
   C. −1
   D. 1
   Answer: A. 2c = 3c + 2 gives c = −2. (Key point: Value of c)

4. With a0 = 5, what is α?
   A. 5
   B. 7
   C. 3
   D. −3
   Answer: B. α − 2 = 5. (Key point: Initial condition)

## Close variants set in other exams

- [Solve aₙ=2aₙ₋₁+2n²](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arecurrence-a-n-2a-2n-squared) — Discrete Structure, chapter 3, asked 6×
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- [Solve aₙ=2aₙ₋₁+3ⁿ](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arecurrence-a-n-2a-3n) — Discrete Structure, chapter 3, asked 6×
- [Solve third-order recurrence with n·4ⁿ](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arecurrence-third-order-n-4-power-n) — Discrete Structure, chapter 3, asked 6×
- [Solve aₙ=5aₙ₋₁−6aₙ₋₂+2ⁿ](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arecurrence-5a-6a-2n-a0-1-a1-4) — Discrete Structure, chapter 3, asked 6×

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