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
Hey MaMNTUNGWA, good to see you again. Step 1: Understand the relationship between square numbers and triangular numbers. A square number S_n can be formed by adding two consecutive triangular numbers, T_n and T_n-1. The formula for the n-th triangular number is T_n = (n(n+1))/(2). Step 2: Identify the triangular numbers for the 10^th square number. The 10^th square number is S_10 = 10^2 = 100. According to the relationship, S_10 = T_10 + T_10-1 = T_10 + T_9. Step 3: Calculate T_9 and T_10. T_9 = (9(9+1))/(2) = (9 × 10)/(2) = (90)/(2) = 45 T_10 = (10(10+1))/(2) = (10 × 11)/(2) = (110)/(2) = 55 To check: T_9 + T_10 = 45 + 55 = 100, which is 10^2. Step 4: Predict the triangular numbers. The triangular numbers used to make the 10^th square number are the 9^th and 10^th triangular numbers. The 9^th triangular number is 45. The 10^th triangular number is 55. Step 5: Draw the pattern to check. To draw the pattern for the 10^th square number (100 dots): • Draw the 10^th triangular number (T_10) as a right-angled triangle of dots. It will have 10 dots in the bottom row, 9 in the row above it, and so on, up to 1 dot at the top. • Draw the 9^th triangular number (T_9) as another right-angled triangle of dots. It will have 9 dots in the bottom row, 8 in the row above it, and so on, up to 1 dot at the top. • Rotate the T_9 pattern by 180^ and place it next to the T_10 pattern. • When combined, the two triangular patterns will perfectly form a 10 × 10 square grid of dots. The triangular numbers used are 45 and 55. Send me the next one 📸