4 WOAKED EXAMPLE 12 Find the value of $$ k $$ in each of these first-order linear recurrence relations, given the first three terms of the sequence. a $$ 350,360,369 \ldots $$ $$ T_{n+1}=0.9 T_{n}+k, T_{1}=350 $$ b $$ 100,80,58.4 \ldots $$ $$ T_{n+1}=1.08 T_{n}+k, T_{1}=100 $$ c $$ 1050,2150,5450 \ldots $$ $$ T_{n+1}=3 T_{n}+k, T_{1}=1050 $$

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**(a) Sequence: $$350, 360, 369, \ldots$$**

The recurrence relation is  
$$
T_{n+1} = 0.9 T_n + k, \quad T_1 = 350.
$$

1. For $$ n = 1 $$, we have:
   $$
   T_2 = 0.9 \cdot 350 + k = 315 + k.
   $$
2. Since $$ T_2 = 360 $$, set up the equation:
   $$
   315 + k = 360.
   $$
3. Solve for $$ k $$:
   $$
   k = 360 - 315 = 45.
   $$
4. **Verification:**  
   Compute $$ T_3 $$:
   $$
   T_3 = 0.9 \cdot 360 + 45 = 324 + 45 = 369.
   $$
   This matches the given sequence.

---

**(b) Sequence: $$100, 80, 58.4, \ldots$$**

The recurrence relation is  
$$
T_{n+1} = 1.08 T_n + k, \quad T_1 = 100.
$$

1. For $$ n = 1 $$, we have:
   $$
   T_2 = 1.08 \cdot 100 + k = 108 + k.
   $$
2. Since $$ T_2 = 80 $$, set up the equation:
   $$
   108 + k = 80.
   $$
3. Solve for $$ k $$:
   $$
   k = 80 - 108 = -28.
   $$
4. **Verification:**  
   Compute $$ T_3 $$:
   $$
   T_3 = 1.08 \cdot 80 - 28 = 86.4 - 28 = 58.4.
   $$
   This is consistent with the given sequence.

---

**(c) Sequence: $$1050, 2150, 5450, \ldots$$**

The recurrence relation is  
$$
T_{n+1} = 3 T_n + k, \quad T_1 = 1050.
$$

1. For $$ n = 1 $$, we have:
   $$
   T_2 = 3 \cdot 1050 + k = 3150 + k.
   $$
2. Since $$ T_2 = 2150 $$, set up the equation:
   $$
   3150 + k = 2150.
   $$
3. Solve for $$ k $$:
   $$
   k = 2150 - 3150 = -1000.
   $$
4. **Verification:**  
   Compute $$ T_3 $$:
   $$
   T_3 = 3 \cdot 2150 - 1000 = 6450 - 1000 = 5450.
   $$
   This confirms the consistency with the given sequence.

---

**Final Answers:**  
(a) $$ k = 45 $$  
(b) $$ k = -28 $$  
(c) $$ k = -1000 $$
$$ k = 45 $$ for part (a), $$ k = -28 $$ for part (b), and $$ k = -1000 $$ for part (c).

4 WOAKED EXAMPLE 12 Find the value of in each of these first-order linear recurrence relations,
given the first three terms of the sequence.
a

b

c

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4 WOAKED EXAMPLE 12 Find the value of in each of these first-order linear recurrence relations, given the first three terms of the sequence. a b c