Question

1. search for patterns in the table. then, use the patterns to complete the next three rows of the table. describe the patterns you used:

table: input column has 1, 2, 3, 4 (and three empty cells); output column has 0, 2, 6, 12 (and three empty cells)

Explanation:

Step1: Analyze the pattern for input

The input values are \(1, 2, 3, 4\), so the next three inputs should be \(5, 6, 7\) (since it's a sequence of consecutive positive integers, increasing by \(1\) each time).

Step2: Analyze the pattern for output

Let's find the relationship between input (\(n\)) and output (\(y\)):

• For \(n = 1\), \(y=0=\frac{1\times(1 - 1)\times(1 + 1)}{2}=\frac{1\times0\times2}{2}=0\)
• For \(n = 2\), \(y = 2=\frac{2\times(2 - 1)\times(2 + 1)}{3}=\frac{2\times1\times3}{3}=2\) (Wait, maybe another way. Let's check the differences between outputs or the formula \(y=\frac{n(n - 1)}{2}\times2\)? Wait, \(n = 1\): \(0\), \(n = 2\): \(2=\frac{2\times(2)}{2}\), \(n=3\): \(6=\frac{3\times4}{2}\), \(n = 4\): \(12=\frac{4\times6}{2}\). Wait, the differences between outputs: \(2-0 = 2\), \(6 - 2=4\), \(12 - 6 = 6\). So the differences are \(2,4,6\) which are \(2\times1,2\times2,2\times3\). So the output for \(n\) is the sum of the first \((n - 1)\) even numbers? Wait, for \(n = 1\), sum of 0 even numbers: \(0\). For \(n = 2\), sum of first 1 even number (\(2\)): \(2\). For \(n = 3\), sum of first 2 even numbers (\(2 + 4\)): \(6\). For \(n = 4\), sum of first 3 even numbers (\(2+4 + 6\)): \(12\). So the formula for output when input is \(n\) is \(y=\sum_{k = 1}^{n - 1}2k=2\times\sum_{k = 1}^{n - 1}k=2\times\frac{(n - 1)n}{2}=n(n - 1)\)? Wait, no:

Wait \(n = 1\): \(1\times0 = 0\) (correct)
\(n = 2\): \(2\times1=2\) (correct)
\(n = 3\): \(3\times2 = 6\) (correct)
\(n = 4\): \(4\times3=12\) (correct). Oh! So the output \(y=n(n - 1)\)

So for \(n = 5\): \(y=5\times4 = 20\)
For \(n = 6\): \(y=6\times5=30\)
For \(n = 7\): \(y=7\times6 = 42\)

Step3: Complete the table

The next three inputs are \(5,6,7\) and the corresponding outputs are \(20,30,42\)

Answer:

The next three rows of the table are:

Input	5	6	7

The pattern for input is consecutive positive integers (increasing by 1). The pattern for output is that the output value is the product of the input value and the input value minus 1 (i.e., \(y = n(n - 1)\) where \(n\) is the input).