Question

mathematics examples:
patterns 1. predict the next pattern.
determine the number of points in the 4th and 5th and nth figure.
conjecture:___________________________

Explanation:

Step1: Analyze the pattern

Looking at the figures:

• Figure 1: 1 point.
• Figure 2: 3 points (1 + 2).
• Figure 3: 6 points (1 + 2 + 3).
• Figure 4: Let's assume the pattern is the sum of the first \(n\) positive integers, where \(n\) is the figure number. The formula for the sum of the first \(n\) positive integers is \(\frac{n(n + 1)}{2}\).

For \(n = 4\): \(\frac{4\times(4 + 1)}{2}=\frac{4\times5}{2}=10\)

Step2: Calculate for Figure 5

Using the same formula \(\frac{n(n + 1)}{2}\) with \(n = 5\):
\(\frac{5\times(5 + 1)}{2}=\frac{5\times6}{2}=15\)

Step3: Calculate for Figure 8

Using the formula \(\frac{n(n + 1)}{2}\) with \(n = 8\):
\(\frac{8\times(8 + 1)}{2}=\frac{8\times9}{2}=36\)

Answer:

• 4th figure: 10 points
• 5th figure: 15 points
• 8th figure: 36 points
• Conjecture: The number of points in the \(n\)-th figure is given by the formula \(\frac{n(n + 1)}{2}\) (the sum of the first \(n\) positive integers).