QUESTION 2
21 Given the quadratic sequence:
2.1.1 Write down the value of the term, , of this sequence.
2.1.2 Delermine an expression for the term of this sequence.
2.1.3 Calculate the number of terms th the sequence.

22 Determine the sum of the whole numbers between 100 and I 000 which are divisible by 11.

Ask by Rose Wright. in South Africa

Jan 14,2025

Question

QUESTION 2
21 Given the quadratic sequence:
2.1.1 Write down the value of the term, , of this sequence.
2.1.2 Delermine an expression for the term of this sequence.
2.1.3 Calculate the number of terms th the sequence.

22 Determine the sum of the whole numbers between 100 and I 000 which are divisible by 11.

Ask by Rose Wright. in South Africa

Jan 14,2025

UpStudy AI · Accepted Answer

Sure, let's tackle each part of your questions step by step.

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### **Question 21**

**Given the quadratic sequence:** $$ 0 ; 5 ; 14 ; \ldots ; 779 ; 860 $$

A **quadratic sequence** has a general term of the form:
$$ a_n = An^2 + Bn + C $$
where $$ A $$, $$ B $$, and $$ C $$ are constants to be determined.

#### **2.1.2. Determine an expression for the $$ n^{\text{th}} $$ term of this sequence.**

**Step 1: Establish Equations Using Given Terms**

Using the first three terms to set up equations:

1. For $$ n = 1 $$: 
   $$ a_1 = A(1)^2 + B(1) + C = 0 $$
   $$ A + B + C = 0 \quad \text{(Equation 1)} $$

2. For $$ n = 2 $$: 
   $$ a_2 = A(2)^2 + B(2) + C = 5 $$
   $$ 4A + 2B + C = 5 \quad \text{(Equation 2)} $$

3. For $$ n = 3 $$: 
   $$ a_3 = A(3)^2 + B(3) + C = 14 $$
   $$ 9A + 3B + C = 14 \quad \text{(Equation 3)} $$

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

Subtract **Equation 1** from **Equation 2**:
$$ (4A + 2B + C) - (A + B + C) = 5 - 0 $$
$$ 3A + B = 5 \quad \text{(Equation 4)} $$

Subtract **Equation 2** from **Equation 3**:
$$ (9A + 3B + C) - (4A + 2B + C) = 14 - 5 $$
$$ 5A + B = 9 \quad \text{(Equation 5)} $$

Subtract **Equation 4** from **Equation 5**:
$$ (5A + B) - (3A + B) = 9 - 5 $$
$$ 2A = 4 $$
$$ A = 2 $$

Plugging $$ A = 2 $$ into **Equation 4**:
$$ 3(2) + B = 5 $$
$$ 6 + B = 5 $$
$$ B = -1 $$

Plugging $$ A = 2 $$ and $$ B = -1 $$ into **Equation 1**:
$$ 2 - 1 + C = 0 $$
$$ C = -1 $$

**Step 3: Write the General Term**

$$ a_n = 2n^2 - n - 1 $$

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#### **2.1.1. Write down the value of the $$ 4^{\text{th}} $$ term, $$ \Gamma_{4} $$, of this sequence.**

Using the general term $$ a_n = 2n^2 - n - 1 $$:

$$ a_4 = 2(4)^2 - 4 - 1 $$
$$ a_4 = 2(16) - 4 - 1 $$
$$ a_4 = 32 - 4 - 1 $$
$$ a_4 = 27 $$

**So, $$ \Gamma_{4} = 27 $$.**

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#### **2.1.3. Calculate the number of terms in the sequence.**

Given the last two terms are 779 and 860.

Set $$ a_k = 779 $$ and $$ a_{k+1} = 860 $$.

Using the general term:
$$ 2k^2 - k - 1 = 779 $$
$$ 2k^2 - k - 780 = 0 $$

**Solve the quadratic equation:**

$$ 2k^2 - k - 780 = 0 $$

Using the quadratic formula $$ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$:

$$ a = 2, \quad b = -1, \quad c = -780 $$
$$ \Delta = (-1)^2 - 4(2)(-780) = 1 + 6240 = 6241 $$
$$ \sqrt{\Delta} = 79 $$
$$ k = \frac{1 \pm 79}{4} $$

Disregarding the negative solution:
$$ k = \frac{1 + 79}{4} = \frac{80}{4} = 20 $$

**Therefore, the number of terms in the sequence is 21** (since $$ a_{k+1} = a_{21} = 860 $$).

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### **Question 22**

**Determine the sum of the whole numbers between 100 and 1,000 which are divisible by 11.**

**Step 1: Identify the First and Last Terms**

- **First multiple of 11 ≥ 100:**
  $$ \frac{100}{11} \approx 9.09 $$
  $$ 11 \times 10 = 110 $$

- **Last multiple of 11 ≤ 1000:**
  $$ \frac{1000}{11} \approx 90.909 $$
  $$ 11 \times 90 = 990 $$

**Sequence of interest:** 110, 121, 132, ..., 990

**Step 2: Determine the Number of Terms ($$ n $$)**

This is an arithmetic sequence where:
- First term ($$ a_1 $$) = 110
- Common difference ($$ d $$) = 11
- Last term ($$ a_n $$) = 990

Using the formula for the $$ n^{\text{th}} $$ term:
$$ a_n = a_1 + (n - 1)d $$
$$ 990 = 110 + (n - 1) \times 11 $$
$$ 990 - 110 = 11(n - 1) $$
$$ 880 = 11(n - 1) $$
$$ n - 1 = 80 $$
$$ n = 81 $$

**Step 3: Calculate the Sum**

Using the formula for the sum of an arithmetic sequence:
$$ S_n = \frac{n}{2} (a_1 + a_n) $$
$$ S_{81} = \frac{81}{2} (110 + 990) $$
$$ S_{81} = 40.5 \times 1100 $$
$$ S_{81} = 44,550 $$

**Therefore, the sum of all whole numbers between 100 and 1,000 divisible by 11 is 44,550.**

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### **Summary of Answers**

- **21.2.1.** The 4th term $$ \Gamma_{4} $$ is **27**.
- **21.2.2.** The general term is $$ a_n = 2n^2 - n - 1 $$.
- **21.2.3.** The sequence contains **21 terms**.
- **22.** The sum of all whole numbers between 100 and 1,000 divisible by 11 is **44,550**.
**Question 21:** - **2.1.1.** The 4th term $$ \Gamma_{4} $$ is **27**. - **2.1.2.** The general term is $$ a_n = 2n^2 - n - 1 $$. - **2.1.3.** The sequence has **21 terms**. **Question 22:** The sum of all whole numbers between 100 and 1,000 divisible by 11 is **44,550**.

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Question 21

2.1.2. Determine an expression for the term of this sequence.

2.1.1. Write down the value of the term, , of this sequence.

2.1.3. Calculate the number of terms in the sequence.

Question 22

Summary of Answers

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