Math Problem Statement

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.)

Solution

Step-by-Step Breakdown

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

  1. For N=0N = 0:
    Sequence: [1][1]

  2. For N=1N = 1:
    Sequence: [−1,0,3][-1, 0, 3]

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

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

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

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

Observing the Pattern:

  • The sequence length increases with NN, starting with 1 element for N=0N = 0, 3 elements for N=1N = 1, 6 elements for N=2N = 2, and so on.
  • The number of terms in each sequence follows the pattern:
    • N=0N = 0: 1 term
    • N=1N = 1: 3 terms
    • N=2N = 2: 6 terms
    • N=3N = 3: 9 terms
    • N=4N = 4: 12 terms
    • N=5N = 5: 15 terms
  • So, the number of terms for NN seems to be 3N+13N + 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=1N = 1, the differences are:

    • 0−(−1)=10 - (-1) = 1
    • 3−0=33 - 0 = 3

  2. For N=2N = 2, the differences are:

    • −3−(−4)=1-3 - (-4) = 1
    • 0−(−3)=30 - (-3) = 3
    • 5−0=55 - 0 = 5
    • 12−5=712 - 5 = 7
    • 21−12=921 - 12 = 9

  3. For N=3N = 3, the differences are:

    • −8−(−9)=1-8 - (-9) = 1
    • −5−(−8)=3-5 - (-8) = 3
    • 0−(−5)=50 - (-5) = 5
    • 7−0=77 - 0 = 7
    • 16−7=916 - 7 = 9
    • 27−16=1127 - 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 a1=−N2a_1 = -N^2.
  2. Each subsequent number is generated by adding an increasing odd number to the previous number.

Algorithm for N=6N = 6:

Following the same pattern:

  1. Start with a1=−62=−36a_1 = -6^2 = -36.
  2. The number of terms will be 3×6+1=193 \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=6N = 6:

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.

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Math Problem Analysis

Mathematical Concepts

Number Sequences
Difference Sequences
Patterns

Formulas

First term: a_1 = -N^2
Next term: a_n = a_(n-1) + (2n - 1)

Theorems

Arithmetic Sequence
Second-Order Difference Sequence

Suitable Grade Level

Grades 10-12

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