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Leaving Cert Applied Maths: Second-order difference equations

How often Second-order difference equations comes up on the Applied Maths papers, every year it was asked, and questions to try.

HL Asked on 3 of the last 3 Higher Level papers, most recently in 2025. banker

OL Asked on 3 of the last 3 Ordinary Level papers, most recently in 2025. banker

Second-order difference equations, Higher Level(7 marks)

Quick ones on Second-order difference equations.

(a)Characteristic equation of Uₙ₊₂ = 5Uₙ₊₁ − 6Uₙ?
  1. x² + 5x − 6 = 0
  2. x² − 5x + 6 = 0
  3. x² − 6x + 5 = 0
(b)Why does a second-order difference equation need two starting values, like U₀ and U₁?
  1. Because the roots must be equal
  2. To find the characteristic roots
  3. To find the two constants α and β
(c)Roots of the characteristic equation of Uₙ₊₂ − Uₙ₊₁ − 2Uₙ = 0?
  1. 2 and −1
  2. 1 and 2
  3. −2 and 1
Show the answers

(a) x² − 5x + 6 = 0

(b) To find the two constants α and β

(c) 2 and −1

Your turn: Higher Level questions on Second-order difference equations.

Higher Level

Asked on 3 of the last 3 Higher Level papers, most recently in 2025. banker

Every paper, year by year

YearWhere it came up
2025Q8, Q10
2024Q9
2023Q6

Links open the State Examinations Commission’s paper for that year.

More Second-order difference equations questions

Second-order difference equations, 2 marks

Uₙ₊₂ = Uₙ₊₁ + Uₙ with U₀ = 1 and U₁ = 1. Find U₅.

  1. 5
  2. 13
  3. 8
Show the answer

8

Each term is the sum of the two before: 1, 1, 2, 3, 5, 8, so U₅ = 8 (the Fibonacci numbers). 5 is U₄ and 13 is U₆; count from U₀.

Second-order difference equations, 2 marks

Characteristic equation of 2Uₙ₊₂ − 7Uₙ₊₁ + 3Uₙ = 0?

  1. 2x² + 7x + 3 = 0
  2. 2x² − 7x + 3 = 0
  3. 3x² − 7x + 2 = 0
Show the answer

2x² − 7x + 3 = 0

Try Uₙ = xⁿ and divide by xⁿ: 2x² − 7x + 3 = 0, keeping each coefficient with its own power. Its roots are 3 and ½.

Second-order difference equations, 3 marks

Uₙ = α(2ⁿ) + β(−1)ⁿ with U₀ = 3 and U₁ = 3. Find α and β.

  1. α = 2, β = 1
  2. α = 1, β = 2
  3. α = 3, β = 0
Show the answer

α = 2, β = 1

n = 0: α + β = 3. n = 1: 2α − β = 3. Adding gives 3α = 6, so α = 2 and β = 1. Check U₁: 4 − 1 = 3.

Ordinary Level

Asked on 3 of the last 3 Ordinary Level papers, most recently in 2025. banker

Every paper, year by year

YearWhere it came up
2025Q10
2024Q10
2023Q8

Links open the State Examinations Commission’s paper for that year.

More Second-order difference equations questions

Second-order difference equations, 2 marks

Characteristic equation of uₙ₊₂ = 5uₙ₊₁ − 6uₙ?

  1. x² + 5x − 6 = 0
  2. x² − 6x + 5 = 0
  3. x² − 5x + 6 = 0
Show the answer

x² − 5x + 6 = 0

Move all terms to one side: uₙ₊₂ − 5uₙ₊₁ + 6uₙ = 0. Replace uₙ₊₂ with x², uₙ₊₁ with x and uₙ with 1.

Second-order difference equations, 2 marks

uₙ₊₂ = uₙ₊₁ + uₙ, u₀ = 1, u₁ = 1. Find u₄.

  1. 8
  2. 5
  3. 3
Show the answer

5

Add the two previous terms: u₂ = 2, u₃ = 3, u₄ = 5. These are the Fibonacci numbers. 8 is u₅.

Second-order difference equations, 3 marks

x² − 5x + 6 = 0 has roots 2 and 3. The general solution uₙ is…

  1. α(2ⁿ) + β(3ⁿ)
  2. 2α + 3β
  3. (α + β)6ⁿ
Show the answer

α(2ⁿ) + β(3ⁿ)

Each root r gives a solution rⁿ, and any mix of the two also works. The constants α and β are then found from two starting values.

Other Applied Maths topics

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