A pattern of hexagons is made using matchsticks. The first hexagon uses 6 matchsticks. Each subsequent hexagon uses 5 more matchsticks than the previous one. How many matchsticks are needed for the 5t

Here's how to solve the problem:

Step 1: Determine the pattern of matchsticks.
Count the matchsticks for the first few hexagons:

• 1st hexagon: 6 matchsticks
• 2nd hexagon: 11 matchsticks
• 3rd hexagon: 16 matchsticks
• 4th hexagon: 21 matchsticks The number of matchsticks forms an arithmetic sequence with the first term $a_1 = 6$ and a common difference $d = 11 - 6 = 5$. The formula for the $n$-th term is $a_n = a_1 + (n-1)d$.

a) How many matches do you need to make the 5th hexagon?
Using the formula for $n=5$:

$$a_5 = 6 + (5-1) \times 5$$ $$a_5 = 6 + 4 \times 5$$ $$a_5 = 6 + 20$$ $$a_5 = 26$$

You need 26 matchsticks for the 5th hexagon.

b) How many matches do you need to make the 6th hexagon?
Using the formula for $n=6$:

$$a_6 = 6 + (6-1) \times 5$$ $$a_6 = 6 + 5 \times 5$$ $$a_6 = 6 + 25$$ $$a_6 = 31$$

You need 31 matchsticks for the 6th hexagon.

c) How many matches do you need to make the 10th hexagon?
Using the formula for $n=10$:

$$a_{10} = 6 + (10-1) \times 5$$ $$a_{10} = 6 + 9 \times 5$$ $$a_{10} = 6 + 45$$ $$a_{10} = 51$$

You need 51 matchsticks for the 10th hexagon.

Final Answers:

a) $\boxed{\text{26 matchsticks}}$

b) $\boxed{\text{31 matchsticks}}$

c) $\boxed{\text{51 matchsticks}}$