Question 3
The numbers in the table below are the result of executing an algorithm that has one parameter N, a non-negative integer, and produces sequences of integers as outputs. For values of N from 0 to 5, the algorithm produces the following sequences of numbers as outputs:


N

output

0


1

-1 0 3

2

-4 -3 0 5 12 21

3

-9 -8 -5 0 7 16 27 40 55

4

-16 -15 -12 -7 0 9 20 33 48 65 84 105

5

-25 -24 -21 -16 -9 0 11 24 39 56 75 96 119 144 171


Determine the algorithm that was used to generate the numbers in this table, and 

Write it down. 

Execute it for N = 6, and write down your result. 

What is the sequence of numbers for N = 6? 

(Give your answer as integers separated by single spaces.)

### Step-by-Step Breakdown

We are tasked with determining the algorithm that generates the sequence based on values of \( N \). Let's analyze the sequences:

1. For \( N = 0 \):  
   Sequence: \( [1] \)

2. For \( N = 1 \):  
   Sequence: \( [-1, 0, 3] \)

3. For \( N = 2 \):  
   Sequence: \( [-4, -3, 0, 5, 12, 21] \)

4. For \( N = 3 \):  
   Sequence: \( [-9, -8, -5, 0, 7, 16, 27, 40, 55] \)

5. For \( N = 4 \):  
   Sequence: \( [-16, -15, -12, -7, 0, 9, 20, 33, 48, 65, 84, 105] \)

6. For \( N = 5 \):  
   Sequence: \( [-25, -24, -21, -16, -9, 0, 11, 24, 39, 56, 75, 96, 119, 144, 171] \)

### Observing the Pattern:
- The sequence length increases with \( N \), starting with 1 element for \( N = 0 \), 3 elements for \( N = 1 \), 6 elements for \( N = 2 \), and so on.
- The number of terms in each sequence follows the pattern:  
  - \( N = 0 \): 1 term
  - \( N = 1 \): 3 terms
  - \( N = 2 \): 6 terms
  - \( N = 3 \): 9 terms
  - \( N = 4 \): 12 terms
  - \( N = 5 \): 15 terms
- So, the number of terms for \( N \) seems to be \( 3N + 1 \).

#### Relationship between terms:
The sequences seem to follow a second-order difference pattern. The differences between consecutive terms can be analyzed:

1. For \( N = 1 \), the differences are:
   - \( 0 - (-1) = 1 \)
   - \( 3 - 0 = 3 \)
   
2. For \( N = 2 \), the differences are:
   - \( -3 - (-4) = 1 \)
   - \( 0 - (-3) = 3 \)
   - \( 5 - 0 = 5 \)
   - \( 12 - 5 = 7 \)
   - \( 21 - 12 = 9 \)
   
3. For \( N = 3 \), the differences are:
   - \( -8 - (-9) = 1 \)
   - \( -5 - (-8) = 3 \)
   - \( 0 - (-5) = 5 \)
   - \( 7 - 0 = 7 \)
   - \( 16 - 7 = 9 \)
   - \( 27 - 16 = 11 \)

This shows a pattern where the differences between terms increase by 2 each time. Specifically, the differences follow the sequence: 1, 3, 5, 7, 9, 11, etc.

### General Algorithm:
1. Start with the first number \( a_1 = -N^2 \).
2. Each subsequent number is generated by adding an increasing odd number to the previous number.

### Algorithm for \( N = 6 \):
Following the same pattern:
1. Start with \( a_1 = -6^2 = -36 \).
2. The number of terms will be \( 3 \times 6 + 1 = 19 \).
3. Generate the terms by adding the differences 1, 3, 5, 7, etc.

Let's compute the sequence.

#### Calculating the sequence for \( N = 6 \):

\[
\begin{aligned}
a_1 &= -36, \\
a_2 &= -36 + 1 = -35, \\
a_3 &= -35 + 3 = -32, \\
a_4 &= -32 + 5 = -27, \\
a_5 &= -27 + 7 = -20, \\
a_6 &= -20 + 9 = -11, \\
a_7 &= -11 + 11 = 0, \\
a_8 &= 0 + 13 = 13, \\
a_9 &= 13 + 15 = 28, \\
a_{10} &= 28 + 17 = 45, \\
a_{11} &= 45 + 19 = 64, \\
a_{12} &= 64 + 21 = 85, \\
a_{13} &= 85 + 23 = 108, \\
a_{14} &= 108 + 25 = 133, \\
a_{15} &= 133 + 27 = 160, \\
a_{16} &= 160 + 29 = 189, \\
a_{17} &= 189 + 31 = 220, \\
a_{18} &= 220 + 33 = 253, \\
a_{19} &= 253 + 35 = 288.
\end{aligned}
\]

Thus, the sequence for \( N = 6 \) is:
\[
-36 -35 -32 -27 -20 -11 0 13 28 45 64 85 108 133 160 189 220 253 288
\]

### Final Answer:
The sequence of numbers for \( N = 6 \) is:
\[
-36 -35 -32 -27 -20 -11 0 13 28 45 64 85 108 133 160 189 220 253 288
\]

Would you like more details, or do you have any further questions?

---

Here are 5 related questions to explore further:
1. What is the general formula for generating terms for any \( N \)?
2. How would the pattern change if the differences increased by 3 instead of 2?
3. Can you express the position of any term in the sequence as a function of \( N \)?
4. What happens to the sequence if you start from a different initial term?
5. How can you modify this sequence to follow a geometric progression instead?

**Tip:** Always look for patterns in differences between terms to discover hidden rules in sequences.

The numbers in the table below are the result of executing an algorithm that has one parameter N, a non-negative integer, and produces sequences of integers as outputs. For values of N from 0 to 5, the algorithm produces the following sequences of numbers as outputs:
N = 0: [1]
N = 1: [-1, 0, 3]
N = 2: [-4, -3, 0, 5, 12, 21]
N = 3: [-9, -8, -5, 0, 7, 16, 27, 40, 55]
N = 4: [-16, -15, -12, -7, 0, 9, 20, 33, 48, 65, 84, 105]
N = 5: [-25, -24, -21, -16, -9, 0, 11, 24, 39, 56, 75, 96, 119, 144, 171]
Determine the algorithm that was used to generate the numbers in this table, and write it down. Execute it for N = 6, and write down your result. What is the sequence of numbers for N = 6? (Give your answer as integers separated by single spaces.)

Discover how to generate a sequence of numbers using an algorithm that takes a non-negative integer N as input. This example covers sequences for N = 0 to N = 5 and explores how the sequence evolves for N = 6. Learn key mathematical concepts such as arithmetic sequences and difference patterns.

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