Miguel would like to figure out how many blocks he would need to build the 25th dragon without actually building it. He knows that there is a pattern to his hexagon dragons. Can you write a rule in words or a math expression that Miguel can use to figure out the total number of blocks for any dragon without having to build it? explain how your rule works.
Sure! To determine the total number of blocks needed to build a specific dragon without actually building it, you can use a mathematical expression based on the given pattern.
The pattern for Miguel's hexagon dragons is that each dragon consists of a central hexagon surrounded by additional layers of hexagons. The number of hexagons in each layer increases by 6 as you move outwards from the center.
To calculate the total number of blocks needed to build the 25th dragon, you can use the following mathematical expression:
Total blocks = 1 + 6 * (1 + 2 + 3 + ... + (n-1))
Let's break down the expression to see how it works:
- The first term, "1," represents the central hexagon.
- "n" represents the number of layers of hexagons in the dragon. In this case, n is 25 for the 25th dragon.
- The term inside the parentheses, "(1 + 2 + 3 + ... + (n-1))," represents the sum of all the numbers from 1 to (n-1).
To find the sum of all the numbers from 1 to (n-1), you can use the formula for the sum of an arithmetic series:
Sum = (n-1) * (n-1 + 1) / 2
Combining these elements, the formula to calculate the total number of blocks needed to build the 25th dragon is:
Total blocks = 1 + 6 * [(n-1) * (n-1 + 1) / 2]
Once you substitute n as 25 and evaluate the expression, you will get the total number of blocks required to build the 25th dragon without actually constructing it.