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
Step 1: Identify the square number and the corresponding triangular numbers. The question asks about the 6^th square number. Following the pattern shown in the examples: • The 2^nd square number (2^2=4) is made from the 1^st and 2^nd triangular numbers (T_1+T_2 = 1+3=4). • The 3^rd square number (3^2=9) is made from the 2^nd and 3^rd triangular numbers (T_2+T_3 = 3+6=9). • The 4^th square number (4^2=16) is made from the 3^rd and 4^th triangular numbers (T_3+T_4 = 6+10=16). • The 5^th square number (5^2=25) is made from the 4^th and 5^th triangular numbers (T_4+T_5 = 10+15=25). Therefore, the 6^th square number (6^2=36) will be made from the 5^th triangular number (T_5) and the 6^th triangular number (T_6). Step 2: Calculate the values of the 5^th and 6^th triangular numbers. The formula for the n^th triangular number is T_n = (n(n+1))/(2). For the 5^th triangular number (T_5): T_5 = (5(5+1))/(2) = (5 × 6)/(2) = (30)/(2) = 15 For the 6^th triangular number (T_6): T_6 = (6(6+1))/(2) = (6 × 7)/(2) = (42)/(2) = 21 Step 3: Show the sum and verify. The sum of the 5^th and 6^th triangular numbers is: T_5 + T_6 = 15 + 21 = 36 This is equal to 6^2, the 6^th square number. Step 4: Describe how to draw the pattern. To draw the pattern for the 6^th square number (36 dots): • Draw the 6^th triangular number (T_6) as a right-angled triangle of dots. This pattern will have 6 dots in the bottom row, 5 in the row above it, and so on, up to 1 dot at the top. • Draw the 5^th triangular number (T_5) as another right-angled triangle of dots. This pattern will have 5 dots in the bottom row, 4 in the row above it, and so on, up to 1 dot at the top. • Rotate the T_5 pattern by 180^ and place it adjacent to the T_6 pattern. The combined patterns will form a 6 × 6 square. The 6^th square number is 36. It is made from the 5^th triangular number (15) and the 6^th triangular number (21). Last free one today — make it count tomorrow, or type /upgrade for unlimited.