Answer
- **Are second‐order linear homogeneous with constant coefficients:**
\(\boxed{\text{(a), (d), (e), and (f).}}\)
- **Are not** (due to non‐constant or non‐linear terms):
\(\boxed{\text{(b) and (c).}}\)
Work Step by Step
A **second‐order linear homogeneous recurrence** with **constant coefficients** has the general form
\[
x_k \;=\; A\,x_{k-1} \;+\; B\,x_{k-2},
\]
where \(A\) and \(B\) are constants (do not depend on \(k\)), and there are no extra (non‐homogeneous) terms such as constants or squares of previous terms.
Let's check each option:
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1. **(a) \(\,a_k = 2\,a_{k-1} - 5\,a_{k-2}\)**
- Order: 2 (depends on \(k-1\) and \(k-2\)).
- Linear: yes, just a linear combination of \(a_{k-1}\) and \(a_{k-2}\).
- Homogeneous: yes, no extra constant or function of \(k\).
- Constant coefficients: yes, \(2\) and \(-5\) are constants.
- **Conclusion:** This **is** a second‐order linear homogeneous recurrence with constant coefficients.
2. **(b) \(\,b_k = k\,b_{k-1} + b_{k-2}\)**
- Order: 2.
- Linear: yes, it is a linear combination.
- Homogeneous: yes, no extra term.
- **But** the coefficient of \(b_{k-1}\) is \(k\), which **depends on \(k\)**, so it is **not** constant.
- **Conclusion:** This is **not** a constant‐coefficient recurrence.
3. **(c) \(\,c_k = 3\,c_{k-1} - c_{k-2}^2\)**
- Order: not truly well‐defined, because we have \(\,c_{k-2}^2\) (a square).
- This is **not linear** (due to the square).
- **Conclusion:** Not a linear recurrence at all.
4. **(d) \(\,d_k = 3\,d_{k-1} + d_{k-2}\)**
- Order: 2.
- Linear: yes.
- Homogeneous: yes, no extra constant term.
- Constant coefficients: yes, \(3\) and \(1\).
- **Conclusion:** This **is** a second‐order linear homogeneous recurrence with constant coefficients.
5. **(e) \(\,r_k = r_{k-1} - r_{k-2}\)**
- Order: 2.
- Linear: yes.
- Homogeneous: yes, no extra term.
- Constant coefficients: yes, coefficients are \(1\) and \(-1\).
- **Conclusion:** This **is** a second‐order linear homogeneous recurrence with constant coefficients.
6. **(f) \(\,s_k = 10\,s_{k-2}\)**
- Order: 2 (it depends on \(k-2\), and implicitly the coefficient of \(s_{k-1}\) is \(0\)).
- Linear: yes, it is \(s_k = 0\cdot s_{k-1} + 10\cdot s_{k-2}\).
- Homogeneous: yes, no extra term.
- Constant coefficients: yes, \(0\) and \(10\).
- **Conclusion:** This **is** a second‐order linear homogeneous recurrence with constant coefficients.
