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

# Solve aₙ=2aₙ₋₁+3ⁿ

Discrete Structure · Chapter 3: Recurrence Relation

- Asked in: 2068 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-2a-3n
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-3

## Question

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

## 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}.

## Answer

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.
```text
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 ✓.

## Check yourself

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

1. Which particular guess?
   A. c·n·3^n
   B. c·2^n
   C. c·n^2
   D. c·3^n, since 3 is not a root
   Answer: D. The root is 2, so there is no duplication. (Key point: Trial form)

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

3. With a1 = 5, what is α?
   A. 2
   B. −2
   C. 5
   D. −5
   Answer: B. 2α + 9 = 5. (Key point: Initial condition)

4. What is a_n?
   A. 3^(n+1) + 2^(n+1)
   B. 3^n − 2^n
   C. 3^(n+1) − 2^(n+1)
   D. 2·3^n
   Answer: C. α = −2 gives −2^(n+1). (Key point: Final answer)

## 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×
- [Solve aₙ=3aₙ₋₁+2ⁿ](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arecurrence-a-n-3a-2n) — Discrete Structure, chapter 3, asked 6×
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- [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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