QUESTION 2 2.1 The arithmetic sequence $$ 4 ; 10 ; 16 ; \ldots $$ is the sequence of the first difference of a quadratic sequence with a first term equal to 3 . Determine the $$ 50^{ ext {th }} $$ term of the quadratic sequence. 2.2 Given the finite arithmetic sequence $$ 5 ; 1 ;-3 ; \ldots-83 ;-87 $$ 2.2.1 Calculate the number of terms in the sequence 2.2.2 Calculate the sum of all the negative numbers in the sequence.

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QUESTION 2 2.1 The arithmetic sequence $$ 4 ; 10 ; 16 ; \ldots $$ is the sequence of the first difference of a quadratic sequence with a first term equal to 3 . Determine the $$ 50^{	ext {th }} $$ term of the quadratic sequence. 2.2 Given the finite arithmetic sequence $$ 5 ; 1 ;-3 ; \ldots-83 ;-87 $$ 2.2.1 Calculate the number of terms in the sequence 2.2.2 Calculate the sum of all the negative numbers in the sequence.

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Let's tackle each part of **Question 2** step by step.

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### **2.1. Determining the $$ 50^{\text{th}} $$ Term of the Quadratic Sequence**

**Given:**
- The first difference sequence is an arithmetic sequence: $$ 4, 10, 16, \ldots $$
- The first term of the quadratic sequence ($$ S_1 $$) is **3**.

**Steps:**

1. **Identify the Common Difference ($$ d $$) of the First Difference Sequence:**
   $$
   d = 10 - 4 = 6
   $$
   
2. **Express the $$ n^{\text{th}} $$ Term of the First Difference Sequence ($$ D_n $$):**
   $$
   D_n = 4 + (n - 1) \times 6 = 6n - 2
   $$
   
3. **Relate the First Difference to the Quadratic Sequence:**
   For a quadratic sequence $$ S_n = an^2 + bn + c $$, the first difference is:
   $$
   S_{n+1} - S_n = 2an + a + b
   $$
   Setting this equal to the first difference sequence:
   $$
   2an + a + b = 6n - 2
   $$
   
4. **Solve for $$ a $$ and $$ b $$:**
   - Compare coefficients:
     $$
     2a = 6 \Rightarrow a = 3
     $$
     $$
     a + b = -2 \Rightarrow 3 + b = -2 \Rightarrow b = -5
     $$
     
5. **Determine the Constant Term ($$ c $$) Using $$ S_1 = 3 $$:**
   $$
   S_1 = 3(1)^2 - 5(1) + c = 3 - 5 + c = 3 \Rightarrow c = 5
   $$
   
6. **Formulate the Quadratic Sequence:**
   $$
   S_n = 3n^2 - 5n + 5
   $$
   
7. **Calculate the $$ 50^{\text{th}} $$ Term ($$ S_{50} $$):**
   $$
   S_{50} = 3(50)^2 - 5(50) + 5 = 3(2500) - 250 + 5 = 7500 - 250 + 5 = 7255
   $$

**Answer:** The $$ 50^{\text{th}} $$ term of the quadratic sequence is **7,255**.

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### **2.2. Working with the Finite Arithmetic Sequence $$ 5, 1, -3, \ldots, -83, -87 $$**

**Given:** 
- First term ($$ a $$) = 5
- Second term = 1 → Common difference ($$ d $$) = $$ 1 - 5 = -4 $$
- Last term ($$ l $$) = -87

#### **2.2.1. Calculating the Number of Terms**

**Formula:**
$$
l = a + (n - 1)d
$$

**Plugging in the values:**
$$
-87 = 5 + (n - 1)(-4)
$$
$$
-87 - 5 = -4(n - 1)
$$
$$
-92 = -4(n - 1)
$$
$$
n - 1 = \frac{-92}{-4} = 23
$$
$$
n = 24
$$

**Answer:** There are **24 terms** in the sequence.

#### **2.2.2. Calculating the Sum of All Negative Numbers in the Sequence**

**Identifying Negative Terms:**
- The first two terms are positive (5 and 1).
- From the 3rd term onwards, the terms are negative.
- Number of negative terms = Total terms - 2 = 24 - 2 = 22

**First Negative Term ($$ S_3 $$):**
$$
S_3 = 5 + (3 - 1)(-4) = 5 - 8 = -3
$$

**Last Term ($$ S_{24} $$):**
$$
S_{24} = -87
$$

**Sum of Negative Terms ($$ S_{\text{neg}} $$):**
$$
S_{\text{neg}} = \frac{\text{Number of terms}}{2} \times (\text{First negative term} + \text{Last term})
$$
$$
S_{\text{neg}} = \frac{22}{2} \times (-3 + (-87)) = 11 \times (-90) = -990
$$

**Answer:** The sum of all negative numbers in the sequence is **−990**.

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### **Final Summary**

- **2.1:** The $$ 50^{\text{th}} $$ term of the quadratic sequence is **7,255**.
- **2.2.1:** There are **24 terms** in the sequence.
- **2.2.2:** The sum of all negative numbers in the sequence is **−990**.
**2.1:** The 50th term of the quadratic sequence is **7,255**. **2.2.1:** There are **24 terms** in the sequence. **2.2.2:** The sum of all negative numbers in the sequence is **−990**.

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