In this pattern, the number of dots in each array is increasing by one in each step. Let's analyze the arrays given:

• The first array has 4 dots.
• The second array has 6 dots.
• The third array has 9 dots.
• The fourth array has 16 dots.

This pattern doesn't follow a simple arithmetic or geometric progression. The differences between consecutive terms are 2, 3, and 7, which makes it more complex. However, based on the numbers, the next array might follow a different rule or might involve some specific reasoning.

Part (a): What will the next array in the pattern look like?

The next logical step seems to be an array where the total number of dots continues to grow significantly. Observing the jump from 9 dots to 16 dots (a difference of 7), the next difference could be even larger. If we follow an increasing difference, the next array might have about 25 dots, potentially forming a 5x5 array.

Part (b): Would a total of 32 dots ever be in an array with this pattern? Explain.

Since the number of dots seems to increase rapidly in the later stages of the pattern, it's possible that 32 dots might appear at some point. However, based on the current observations, the jump between arrays is increasing by an unpredictable amount. If we continue this pattern, it seems possible but would require checking future steps of the pattern.

Do you want further details or have any questions?

Relative Questions:

1. What is the rule governing the jump in the number of dots between arrays?
2. How can we predict the number of dots in future arrays with an unknown pattern?
3. Can the pattern of dots be modeled with a mathematical function or formula?
4. Is there any relationship between the shape of the array and the number of dots it contains?
5. How does this dot pattern compare with common sequences like square or triangular numbers?

Tip:

When dealing with number patterns, always check if the differences between terms are increasing or following another sequence themselves, like triangular numbers.