This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
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) × 5 a_5 = 6 + 4 × 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) × 5 a_6 = 6 + 5 × 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) × 5 a_10 = 6 + 9 × 5 a_10 = 6 + 45 a_10 = 51 You need 51 matchsticks for the 10th hexagon. Final Answers: a) 26 matchsticks b) 31 matchsticks c) 51 matchsticks Send me the next one 📸