Mavhemaficu Grinde 12 7 TFeornary 1025 QUESTION 1 Investigate the relationship between the common difference, $$ d $$, and the Difference, $$ D $$, between (the prodnet of the first and the third temis) and (the squme of the middle term) of any tree consecutive numbers of a linear sequence. NOTV: - If the sequence is given by $$ a ; b ; c, \ldots $$ the Difference $$ (D)=b^{1}-a imes c $$ Step I - Use FIVE different linear sequerices with common difference, $$ d $$, of 1 and calculale the difference, Dess stated above. (4) Step 2 - Write doun your conjecture in surds. (2) Step 3 - Repeal step 1: chatge the common difference, d between the terms to 2 . (4) Step 4 - Repeat step 1: change the common difference, d between the termis to 3 . (4) Step 5 - Repeat step 1' change the common difference, $$ d $$ between the terms to a number of your choioz (NOT 1, 2 or 3 ). (4) Step 6 - Write down your conjecture in words (2) Step 7 - Is this true in general? Prove your conjecture. (5) Step 8 - Overall presentation and variety of sequences used. (10) [35]

Question

Mavhemaficu Grinde 12 7 TFeornary 1025 QUESTION 1 Investigate the relationship between the common difference, $$ d $$, and the Difference, $$ D $$, between (the prodnet of the first and the third temis) and (the squme of the middle term) of any tree consecutive numbers of a linear sequence. NOTV: - If the sequence is given by $$ a ; b ; c, \ldots $$ the Difference $$ (D)=b^{1}-a 	imes c $$ Step I - Use FIVE different linear sequerices with common difference, $$ d $$, of 1 and calculale the difference, Dess stated above. (4) Step 2 - Write doun your conjecture in surds. (2) Step 3 - Repeal step 1: chatge the common difference, d between the terms to 2 . (4) Step 4 - Repeat step 1: change the common difference, d between the termis to 3 . (4) Step 5 - Repeat step 1' change the common difference, $$ d $$ between the terms to a number of your choioz (NOT 1, 2 or 3 ). (4) Step 6 - Write down your conjecture in words (2) Step 7 - Is this true in general? Prove your conjecture. (5) Step 8 - Overall presentation and variety of sequences used. (10) [35]

UpStudy AI · Accepted Answer

To solve the problem, we will follow the steps outlined in the question. We will investigate the relationship between the common difference $$ d $$ and the difference $$ D $$ for different linear sequences.

### Step 1: Common Difference $$ d = 1 $$

Let's consider five different linear sequences with a common difference of $$ d = 1 $$:

1. Sequence: $$ 1, 2, 3 $$
   - $$ a = 1, b = 2, c = 3 $$
   - $$ D = b^2 - a \times c = 2^2 - 1 \times 3 = 4 - 3 = 1 $$

2. Sequence: $$ 2, 3, 4 $$
   - $$ a = 2, b = 3, c = 4 $$
   - $$ D = 3^2 - 2 \times 4 = 9 - 8 = 1 $$

3. Sequence: $$ 3, 4, 5 $$
   - $$ a = 3, b = 4, c = 5 $$
   - $$ D = 4^2 - 3 \times 5 = 16 - 15 = 1 $$

4. Sequence: $$ 4, 5, 6 $$
   - $$ a = 4, b = 5, c = 6 $$
   - $$ D = 5^2 - 4 \times 6 = 25 - 24 = 1 $$

5. Sequence: $$ 5, 6, 7 $$
   - $$ a = 5, b = 6, c = 7 $$
   - $$ D = 6^2 - 5 \times 7 = 36 - 35 = 1 $$

**Results for $$ d = 1 $$:**
- The difference $$ D $$ is consistently $$ 1 $$.

### Step 2: Conjecture in Surds

Based on the results from Step 1, we can conjecture that:
$$
D = 1 \quad \text{for } d = 1
$$

### Step 3: Common Difference $$ d = 2 $$

Now, let's change the common difference to $$ d = 2 $$ and analyze five sequences:

1. Sequence: $$ 1, 3, 5 $$
   - $$ a = 1, b = 3, c = 5 $$
   - $$ D = 3^2 - 1 \times 5 = 9 - 5 = 4 $$

2. Sequence: $$ 2, 4, 6 $$
   - $$ a = 2, b = 4, c = 6 $$
   - $$ D = 4^2 - 2 \times 6 = 16 - 12 = 4 $$

3. Sequence: $$ 3, 5, 7 $$
   - $$ a = 3, b = 5, c = 7 $$
   - $$ D = 5^2 - 3 \times 7 = 25 - 21 = 4 $$

4. Sequence: $$ 4, 6, 8 $$
   - $$ a = 4, b = 6, c = 8 $$
   - $$ D = 6^2 - 4 \times 8 = 36 - 32 = 4 $$

5. Sequence: $$ 5, 7, 9 $$
   - $$ a = 5, b = 7, c = 9 $$
   - $$ D = 7^2 - 5 \times 9 = 49 - 45 = 4 $$

**Results for $$ d = 2 $$:**
- The difference $$ D $$ is consistently $$ 4 $$.

### Step 4: Conjecture in Surds

Based on the results from Step 3, we can conjecture that:
$$
D = 4 \quad \text{for } d = 2
$$

### Step 5: Common Difference $$ d = 3 $$

Now, let's change the common difference to $$ d = 3 $$:

1. Sequence: $$ 1, 4, 7 $$
   - $$ a = 1, b = 4, c = 7 $$
   - $$ D = 4^2 - 1 \times 7 = 16 - 7 = 9 $$

2. Sequence: $$ 2, 5, 8 $$
   - $$ a = 2, b = 5, c = 8 $$
   - $$ D = 5^2 - 2 \times 8 = 25 - 16 = 9 $$

3. Sequence: $$ 3, 6, 9 $$
   - $$ a = 3, b = 6, c = 9 $$
   - $$ D = 6^2 - 3 \times 9 = 36 - 27 = 9 $$

4. Sequence: $$ 4, 7, 10 $$
   - $$ a = 4, b = 7, c = 10 $$
   - $$ D = 7^2 - 4 \times 10 = 49 - 40 = 9 $$

5. Sequence: $$ 5, 8, 11 $$
   - $$ a = 5, b = 8, c = 11 $$
   - $$ D = 8^2 - 5 \times 11 = 64 - 55 = 9 $$

**Results for $$ d = 3 $$:**
- The difference $$ D $$ is consistently $$ 9 $$.

### Step 6: Conjecture in Words

Based on the results from Steps 1, 3, and 5, we can state:
- The difference $$ D $$ is equal to $$ d^2 $$ for any common difference $$ d $$.

### Step 7: General Proof of the Conjecture

To prove the conjecture, we can express the difference $$ D $$ mathematically:

Given three consecutive terms of a linear sequence:
- $$ a $$ (first term)
- $$ b = a + d $$ (second term)
- $$ c = a + 2d $$ (third term)

The difference $$ D $$ can be expressed as:
$$
D = b^2 - a \times c
$$
Substituting $$ b $$ and $$ c $$:
$$
D = (a + d)^2 - a(a + 2d)
$$
Expanding both terms:
$$
D = (a^2 + 2ad + d^2) - (a^2 + 2ad) = d^2
$$
Thus, we have shown that $$ D = d^2 $$ for any common difference $$ d $$.

### Step 8: Overall Presentation and Variety of Sequences Used

In this investigation, we have
For any linear sequence with a common difference $$ d $$, the difference $$ D $$ between the square of the middle term and the product of the first and third terms is always $$ D = d^2 $$.

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