Lesson test: Linear recurrences

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Linear recurrences

1.[2p]
Solve $a_{n} = 7 a_{n - 1} - 10 a_{n - 2}$ with $a_{0} = 2$ and $a_{1} = 7$ . What is $a_{6}$ ?

CorrectNot quite: 15689

2.[1p]
What is the larger root of the characteristic equation of $a_{n} = a_{n - 1} + 12 a_{n - 2}$ ?

CorrectNot quite: 4

3.[2p]
The Lucas numbers obey $L_{n} = L_{n - 1} + L_{n - 2}$ with $L_{0} = 2$ and $L_{1} = 1$ . With $φ = \frac{1 + \sqrt{5}}{2}$ and $ψ = \frac{1 - \sqrt{5}}{2}$ , which formula gives $L_{n}$ ?

$φ^{n} + ψ^{n}$

Correct

$\frac{φ^{n} - ψ^{n}}{\sqrt{5}}$

The answer is: $\varphi^n + \psi^n$

$\frac{φ^{n} + ψ^{n}}{\sqrt{5}}$

The answer is: $\varphi^n + \psi^n$

$2 φ^{n} - ψ^{n}$

The answer is: $\varphi^n + \psi^n$

4.[2p]
For $n \geq 0$ , what is the largest possible difference between $\frac{φ^{n}}{\sqrt{5}}$ and $F_{n}$ , the bound that makes $F_{n}$ its nearest integer? Give it to three decimal places.

CorrectNot quite: 0.447

5.[2p]
Solve $a_{n} = 8 a_{n - 1} - 16 a_{n - 2}$ with $a_{0} = 1$ and $a_{1} = 8$ . What is $a_{5}$ ?

CorrectNot quite: 6144

6.[2p]
Solve $a_{n} = 3 a_{n - 1} + 4$ with $a_{0} = 0$ . What is $a_{8}$ ?

CorrectNot quite: 13120

7.[2p]
Solve $a_{n} = a_{n - 1} + 2 n$ with $a_{0} = 5$ . What is $a_{20}$ ?

CorrectNot quite: 425

8.[3p]
Match each recurrence to the form of its general solution.

$a_{n} = 5 a_{n - 1} - 6 a_{n - 2}$
$a_{n} = 4 a_{n - 1} - 4 a_{n - 2}$
$a_{n} = a_{n - 1} + 2 a_{n - 2}$
$a_{n} = 2 a_{n - 1} + 1$

$(A + B n) 2^{n}$
$A \times 2^{n} - 1$
$A \times 2^{n} + B (- 1)^{n}$
$A \times 2^{n} + B \times 3^{n}$

Show the answer

$a_{n} = 5 a_{n - 1} - 6 a_{n - 2}$ : $A \times 2^{n} + B \times 3^{n}$ $a_{n} = 4 a_{n - 1} - 4 a_{n - 2}$ : $(A + B n) 2^{n}$ $a_{n} = a_{n - 1} + 2 a_{n - 2}$ : $A \times 2^{n} + B (- 1)^{n}$ $a_{n} = 2 a_{n - 1} + 1$ : $A \times 2^{n} - 1$

9.[3p]
Which of these statements are correct?

Select all that apply

If $x_{n}$ and $y_{n}$ both satisfy a linear homogeneous recurrence with constant coefficients, so does $x_{n} + y_{n}$

Correct

If $r$ is a double root of the characteristic equation, both $r^{n}$ and $n r^{n}$ satisfy the recurrence

Correct

Every solution of a recurrence with a forcing term is one particular solution plus a solution of the homogeneous recurrence

Correct

Binet's formula gives an integer only when $n$ is even

The answer is: If $x_n$ and $y_n$ both satisfy a linear homogeneous recurrence with constant coefficients, so does $x_n + y_n$, If $r$ is a double root of the characteristic equation, both $r^n$ and $nr^n$ satisfy the recurrence, Every solution of a recurrence with a forcing term is one particular solution plus a solution of the homogeneous recurrence

The recurrence $a_{n} = a_{n - 1} + 3$ has a constant particular solution

The characteristic equation solves $D_{n} = (n - 1) (D_{n - 1} + D_{n - 2})$ , since that recurrence is linear

1.[2p] Solve an=7an-1-10an-2 with a0=2 and a1=7. What is a6?

2.[1p] What is the larger root of the characteristic equation of an=an-1+12an-2?

3.[2p] The Lucas numbers obey Ln=Ln-1+Ln-2 with L0=2 and L1=1. With φ=1+52 and ψ=1-52, which formula gives Ln?

4.[2p] For n≥0, what is the largest possible difference between φn5 and Fn, the bound that makes Fn its nearest integer? Give it to three decimal places.

5.[2p] Solve an=8an-1-16an-2 with a0=1 and a1=8. What is a5?

6.[2p] Solve an=3an-1+4 with a0=0. What is a8?

7.[2p] Solve an=an-1+2n with a0=5. What is a20?

8.[3p] Match each recurrence to the form of its general solution.

9.[3p] Which of these statements are correct?

1.[2p]
Solve $a_{n} = 7 a_{n - 1} - 10 a_{n - 2}$ with $a_{0} = 2$ and $a_{1} = 7$ . What is $a_{6}$ ?

2.[1p]
What is the larger root of the characteristic equation of $a_{n} = a_{n - 1} + 12 a_{n - 2}$ ?

3.[2p]
The Lucas numbers obey $L_{n} = L_{n - 1} + L_{n - 2}$ with $L_{0} = 2$ and $L_{1} = 1$ . With $φ = \frac{1 + \sqrt{5}}{2}$ and $ψ = \frac{1 - \sqrt{5}}{2}$ , which formula gives $L_{n}$ ?

4.[2p]
For $n \geq 0$ , what is the largest possible difference between $\frac{φ^{n}}{\sqrt{5}}$ and $F_{n}$ , the bound that makes $F_{n}$ its nearest integer? Give it to three decimal places.

5.[2p]
Solve $a_{n} = 8 a_{n - 1} - 16 a_{n - 2}$ with $a_{0} = 1$ and $a_{1} = 8$ . What is $a_{5}$ ?

6.[2p]
Solve $a_{n} = 3 a_{n - 1} + 4$ with $a_{0} = 0$ . What is $a_{8}$ ?

7.[2p]
Solve $a_{n} = a_{n - 1} + 2 n$ with $a_{0} = 5$ . What is $a_{20}$ ?

8.[3p]
Match each recurrence to the form of its general solution.

9.[3p]
Which of these statements are correct?