Given the quadratic sequence: $$ -5,12,27 $$ 2.1 Determine the next two terms of the quadratic sequence $$ \quad(2) $$ 2.2 Determine the expression for the general term of the quadratic sequence 2.3 Which term(s) of the quadratic sequence has a value of 51 ? (3)

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Certainly! Let's analyze the given quadratic sequence step by step.

### Given Sequence:
$$
-5, \quad 12, \quad 27
$$

### 2.1 Determine the Next Two Terms of the Quadratic Sequence

**Method: Using the General Form of a Quadratic Sequence**

A quadratic sequence can be represented by the general term:
$$
a_n = An^2 + Bn + C
$$
where:
- $$ A $$, $$ B $$, and $$ C $$ are constants.
- $$ n $$ is the term number.

**Step 1: Set Up Equations Using Known Terms**

Given the first three terms:
1. For $$ n = 1 $$: 
   $$
   A(1)^2 + B(1) + C = -5 \quad \Rightarrow \quad A + B + C = -5 \quad \text{(Equation 1)}
   $$
2. For $$ n = 2 $$: 
   $$
   A(2)^2 + B(2) + C = 12 \quad \Rightarrow \quad 4A + 2B + C = 12 \quad \text{(Equation 2)}
   $$
3. For $$ n = 3 $$: 
   $$
   A(3)^2 + B(3) + C = 27 \quad \Rightarrow \quad 9A + 3B + C = 27 \quad \text{(Equation 3)}
   $$

**Step 2: Solve the System of Equations**

Subtract **Equation 1** from **Equation 2**:
$$
(4A + 2B + C) - (A + B + C) = 12 - (-5) \
3A + B = 17 \quad \text{(Equation 4)}
$$

Subtract **Equation 2** from **Equation 3**:
$$
(9A + 3B + C) - (4A + 2B + C) = 27 - 12 \
5A + B = 15 \quad \text{(Equation 5)}
$$

Subtract **Equation 4** from **Equation 5**:
$$
(5A + B) - (3A + B) = 15 - 17 \
2A = -2 \quad \Rightarrow \quad A = -1
$$

Plug $$ A = -1 $$ into **Equation 4**:
$$
3(-1) + B = 17 \
-3 + B = 17 \quad \Rightarrow \quad B = 20
$$

Plug $$ A = -1 $$ and $$ B = 20 $$ into **Equation 1**:
$$
-1 + 20 + C = -5 \
19 + C = -5 \quad \Rightarrow \quad C = -24
$$

**Step 3: Formulate the General Term**

$$
a_n = -n^2 + 20n - 24
$$

**Step 4: Calculate the Next Two Terms ($$ n = 4 $$ and $$ n = 5 $$)**

1. For $$ n = 4 $$:
   $$
   a_4 = -4^2 + 20(4) - 24 = -16 + 80 - 24 = 40
   $$
2. For $$ n = 5 $$:
   $$
   a_5 = -5^2 + 20(5) - 24 = -25 + 100 - 24 = 51
   $$

**Next Two Terms:** **40** and **51**

---

### 2.2 Determine the Expression for the General Term of the Quadratic Sequence

From the steps above, the general term $$ a_n $$ of the quadratic sequence is:
$$
a_n = -n^2 + 20n - 24
$$

---

### 2.3 Which Term(s) of the Quadratic Sequence Has a Value of 51?

**Given:**
$$
a_n = 51
$$
**Find:** $$ n $$ such that:
$$
-n^2 + 20n - 24 = 51
$$

**Step 1: Set Up the Equation**
$$
-n^2 + 20n - 24 = 51
$$

**Step 2: Rearrange to Standard Quadratic Form**
$$
-n^2 + 20n - 75 = 0 \
n^2 - 20n + 75 = 0 \quad \text{(Multiplying both sides by -1)}
$$

**Step 3: Solve the Quadratic Equation**

Use the quadratic formula:
$$
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
where $$ a = 1 $$, $$ b = -20 $$, and $$ c = 75 $$.

$$
n = \frac{20 \pm \sqrt{(-20)^2 - 4(1)(75)}}{2(1)} \
n = \frac{20 \pm \sqrt{400 - 300}}{2} \
n = \frac{20 \pm \sqrt{100}}{2} \
n = \frac{20 \pm 10}{2}
$$

**Solutions:**
1. $$ n = \frac{20 + 10}{2} = 15 $$
2. $$ n = \frac{20 - 10}{2} = 5 $$

**Verification:**

From **2.1**, we already calculated $$ a_5 = 51 $$.

For $$ n = 15 $$:
$$
a_{15} = -15^2 + 20(15) - 24 = -225 + 300 - 24 = 51
$$

**Conclusion:** The terms at positions $$ n = 5 $$ and $$ n = 15 $$ have the value **51**.
**2.1 Next Two Terms:** 40 and 51 **2.2 General Term:** $$ a_n = -n^2 + 20n - 24 $$ **2.3 Terms with Value 51:** 5th and 15th terms

Given the quadratic sequence: \( -5,12,27 \) 2.1 Determine the next two terms of the quadratic sequence \( \quad(2) \) 2.2 Determine the expression for the general term of the quadratic sequence 2.3 Which term(s) of the quadratic sequence has a value of 51 ? (3)

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