Investigate the reiationship between the common differcace, $$ d $$, and the Difference, $$ D $$, between (the produet of the first and the third termis) and (the square of the middle term) of any three consecutive cumbers of a linear sequence. NOTE: - If the sequence is given by $$ a ; b ; c $$, .. the Difference $$ (D)=b^{2}-a imes c $$ Step 1 - Use FIVE differect linear sequences with common difference, $$ d $$, of $$ I $$ and calculate the diffirence, $$ D $$ a 5 stated above. (4) Step 2 - Write dōw your conjecture in words. (2) Step 3 - Repeat şep 1: change the common difference, d between the terms to 2 . (4) Step 4 - Kepeat step 1. change the common differcace, d between the terms to 3 . (4) Step 5 - Repeat step ! change the common difference, $$ d $$ berween the terms to a number of your choiec (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) [39]

Question

Investigate the reiationship between the common differcace, $$ d $$, and the Difference, $$ D $$, between (the produet of the first and the third termis) and (the square of the middle term) of any three consecutive cumbers of a linear sequence. NOTE: - If the sequence is given by $$ a ; b ; c $$, .. the Difference $$ (D)=b^{2}-a 	imes c $$ Step 1 - Use FIVE differect linear sequences with common difference, $$ d $$, of $$ I $$ and calculate the diffirence, $$ D $$ a 5 stated above. (4) Step 2 - Write dōw your conjecture in words. (2) Step 3 - Repeat şep 1: change the common difference, d between the terms to 2 . (4) Step 4 - Kepeat step 1. change the common differcace, d between the terms to 3 . (4) Step 5 - Repeat step ! change the common difference, $$ d $$ berween the terms to a number of your choiec (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) [39]

UpStudy AI · Accepted Answer

To investigate the relationship between the common difference $$ d $$ and the difference $$ D $$ defined as $$ D = b^2 - a 	imes c $$ for three consecutive terms of a linear sequence, we will follow the steps outlined in the problem.

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

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

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

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

3. Sequence: $$ 7, 8, 9 $$
   - $$ a = 7, b = 8, c = 9 $$
   - $$ D = 8^2 - 7 	imes 9 = 64 - 63 = 1 $$

4. Sequence: $$ 10, 11, 12 $$
   - $$ a = 10, b = 11, c = 12 $$
   - $$ D = 11^2 - 10 	imes 12 = 121 - 120 = 1 $$

5. Sequence: $$ 13, 14, 15 $$
   - $$ a = 13, b = 14, c = 15 $$
   - $$ D = 14^2 - 13 	imes 15 = 196 - 195 = 1 $$

### Step 2: Conjecture for $$ d = 1 $$

For a linear sequence with a common difference of $$ 1 $$, the difference $$ D $$ between the product of the first and third terms and the square of the middle term is always $$ 1 $$.

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

Now, let's consider five different linear sequences with a common difference of $$ d = 2 $$:

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

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

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

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

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

### Step 4: Conjecture for $$ d = 2 $$

For a linear sequence with a common difference of $$ 2 $$, the difference $$ D $$ is always $$ 4 $$.

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

Now, let's consider five different linear sequences with a common difference of $$ d = 3 $$:

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

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

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

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

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

### Step 6: Conjecture for $$ d = 3 $$

For a linear sequence with a common difference of $$ 3 $$, the difference $$ D $$ is always $$ 9 $$.

### Step 7: Common Difference $$ d = 4 $$

Now, let's consider five different linear sequences with a common difference of $$ d = 4 $$:

1. Sequence: $$ 1, 5, 9 $$
   - $$ a = 1, b = 5, c = 9 $$
   - $$ D = 5^2 - 1 	imes 9 = 25 - 9 = 16 $$

2. Sequence: $$ 2, 6, 10 $$
   - $$ a = 2, b = 6, c = 10 $$
   - $$ D = 6^2 - 2 	imes 10 = 36 - 20 = 16 $$

3. Sequence: $$ 3, 7, 11 $$
   - $$ a = 3, b = 7, c = 11 $$
   - $$ D = 7^2 - 3 	imes 11 = 49 - 33 = 16 $$

4. Sequence: $$ 4, 8, 12 $$
   - $$ a = 4, b = 8, c = 12 $$
   - $$ D = 8^2 - 4 	imes 12 = 64 - 48 = 16 $$

5. Sequence: $$ 5,
For any three consecutive terms in a linear sequence with a common difference $$ d $$, the difference $$ D $$ between the square of the middle term and the product of the first and third terms is always $$ d^2 $$.

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