---
title: "Solve aₙ=2aₙ₋₁+2ⁿ"
subject: "Discrete Structure"
chapter: 3
key: "semester_4/discrete_structure:recurrence-a-n-2a-2n-a0-2"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/recurrence-a-n-2a-2n-a0-2
site: BCT Engineering
---

# Solve aₙ=2aₙ₋₁+2ⁿ

Discrete Structure · Chapter 3: Recurrence Relation

- Asked in: 2070 Magh, 8 marks
- Video: [aₙ = 2aₙ₋₁ + 2ⁿ ,a(0)=2 | Non-Homogenous Recurrence Relation |Unit-2  Discrete Mathematics | MA3354](https://www.youtube.com/watch?v=8nyMiyEXiyk) — TechSensei
- 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-2n-a0-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-3

## Question

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

## Short answer

Homogeneous part α·2^n. Since 2 is a root, particular solution is n·2^n. General solution (α+n)·2^n; a_0=2 gives α=2, so a_n = (n+2)·2^n.

## Answer

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

#### Step 1. Homogeneous part
aₙ = 2aₙ₋₁ gives r = 2, so aₙ⁽ʰ⁾ = α·2ⁿ.

#### Step 2. Particular solution
F(n) = 2ⁿ and 2 **is** a root of the characteristic equation (multiplicity 1), so the guess c·2ⁿ would just duplicate the homogeneous solution. Use aₙ⁽ᵖ⁾ = c·n·2ⁿ.
```text
c n 2ⁿ = 2c(n-1)2ⁿ⁻¹ + 2ⁿ
c n 2ⁿ = c(n-1)2ⁿ + 2ⁿ        (divide by 2ⁿ)
c n    = c n - c + 1
c = 1
```
aₙ⁽ᵖ⁾ = n·2ⁿ.

#### Step 3. All solutions
aₙ = α·2ⁿ + n·2ⁿ = (α + n)2ⁿ.

#### Step 4. Initial condition a₀ = 2
α = 2.

**Answer: aₙ = (n + 2)·2ⁿ**

Check: a₁ = 3·2 = 6 = 2(2) + 2 ✓; a₂ = 4·4 = 16 = 2(6) + 4 ✓; a₃ = 5·8 = 40 = 2(16) + 8 ✓.

## Check yourself

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

1. Why is c·n·2^n used?
   A. 2^n is a polynomial
   B. The initial condition is 2
   C. 2 is a root of the characteristic equation, so c·2^n duplicates the homogeneous solution
   D. Because the relation is order 2
   Answer: C. The guess must be independent of α·2^n. (Key point: Why modify guess)

2. What is c?
   A. −1
   B. 2
   C. 0
   D. 1
   Answer: D. cn = cn − c + 1 gives c = 1. (Key point: Value of c)

3. What is the general solution?
   A. a_n = (α + n)·2^n
   B. a_n = α·2^n + 2^n
   C. a_n = α·2^n
   D. a_n = α·n·2^n
   Answer: A. Homogeneous plus n·2^n. (Key point: General solution)

4. With a0 = 2, what is a_n?
   A. (n + 1)·2^n
   B. 2^(n+1)
   C. (n + 2)·2^n
   D. n·2^n
   Answer: C. α = 2. (Key point: Final answer)

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