MTH401

1. What is a homogeneous recurrence relation?

a) A recurrence relation where each term is a constant multiple of the previous one b) A recurrence relation with no constant term and only terms involving previous values c) A recurrence relation with both constant and non-constant terms d) A recurrence relation involving only arithmetic sequences

2. What is the general form of a linear homogeneous recurrence relation of order $n$ ?

a)

a_{n} = c_{1} a_{n - 1} + c_{2} a_{n - 2} + \dots + c_{k} a_{n - k} + f (n)

b)

a_{n} = c_{1} a_{n - 1} + c_{2} a_{n - 2} + \dots + c_{k} a_{n - k}

c)

a_{n} = c_{1} a_{n - 1} + c_{2} a_{n - 2}

d)

a_{n} = a_{n - 1} + a_{n - 2} + f (n)

3. The characteristic equation of the recurrence relation $a_{n} - 5 a_{n - 1} + 6 a_{n - 2} = 0$ is:

a)

r^{2} - 5 r + 6 = 0

b)

r^{2} + 5 r + 6 = 0

c)

r^{2} - 6 r + 5 = 0

d)

r^{2} + 6 r + 5 = 0

4. If the characteristic equation has distinct real roots, the general solution is given by:

a)

a_{n} = C_{1} r_{1}^{n} + C_{2} r_{2}^{n} + \dots + C_{k} r_{k}^{n}

b)

a_{n} = C_{1} n r_{n}^{n} + C_{2} r_{n}^{n}

c)

a_{n} = C_{1} + C_{2} n

d)

a_{n} = C_{1} e^{r_{1} n} + C_{2} e^{r_{2} n}

5. If a characteristic equation has a repeated root $r$ of multiplicity $m$ , the general solution includes:

a)

C_{1} r_{n}^{n} + C_{2} r_{n}^{n} + \dots + C_{m} r_{n}^{n}

b)

C_{1} r_{n}^{n} + C_{2} n r_{n}^{n} + C_{3} n^{2} r_{n}^{n} + \dots + C_{m} n^{m - 1} r_{n}^{n}

c)

C_{1} r_{n}^{n} + C_{2} n^{m} r_{n}^{n}

d)

C_{1} e^{r n} + C_{2} e^{- r n}

6. The recurrence relation $a_{n} - 4 a_{n - 1} + 4 a_{n - 2} = 0$ has characteristic roots:

a)

r = 2, 2

b)

r = 4, - 4

c)

r = 4, 4

d)

r = 2, - 2

7. If the characteristic equation has complex roots $r = α \pm β i$ , the general solution contains terms of the form:

a)

C_{1} e^{(α + β i) n} + C_{2} e^{(α - β i) n}

b)

C_{1} {(α + β i)}^{n} + C_{2} {(α - β i)}^{n}

c)

C_{1} e^{α n} \cos (β n) + C_{2} e^{α n} \sin (β n)

d)

C_{1} e^{α n} + C_{2} e^{- α n}

8. What is the characteristic equation of the recurrence relation $a_{n} - 3 a_{n - 1} + 2 a_{n - 2} = 0$ ?

a)

r^{2} - 3 r + 2 = 0

b)

r^{2} + 3 r + 2 = 0

c)

r^{2} - 2 r + 3 = 0

d)

r^{2} + 2 r + 3 = 0

9. The inverse operator method is used to solve recurrence relations by treating them as:

a) Differential equations b) Polynomial equations c) Operator equations d) Algebraic equations

10. If a recurrence relation is expressed as $L (a_{n}) = f (n)$ , what does $L$ represent?

a) A differential operator b) A recurrence operator c) A characteristic polynomial d) A Laplace transform

11. The inverse operator $L^{- 1}$ is used to:

a) Find the characteristic roots b) Determine the general solution c) Compute eigenvalues d) Solve for the particular solution

12. The operator $E$ in recurrence relations represents:

a) Differentiation b) Integration c) The shift operator d) The inverse Laplace transform

13. For the recurrence relation $a_{n} - 2 a_{n - 1} + a_{n - 2} = n^{2}$ , what type of function does the particular solution involve?

a) Exponential b) Polynomial c) Trigonometric d) Logarithmic

14. If the recurrence relation has a non-homogeneous term $f (n) = 3^{n}$ , the form of the particular solution is usually:

a)

A 3^{n}

b)

A n 3^{n}

c)

A n^{2} 3^{n}

d)

A e^{n}

15. The method of inverse operators is useful when:

a) The roots of the characteristic equation are real and distinct b) The recurrence relation has a non-homogeneous part c) The recurrence relation is purely homogeneous d) The equation is first-order only

Homogeneous Recurrence Relations MCQs

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