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.
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Here are the solutions to the questions. 1a) A 2D shape is a flat figure that has only two dimensions: length and width. It has no depth or thickness. 1b) Three examples of 2D shapes are: • Triangle • Square • Circle 1c) Features of 2D shapes include: • They are flat. • They have length and width (or height). • They have no depth or thickness. • They can be measured in terms of area and perimeter. 1d) 2D shapes differ from 3D shapes because 2D shapes are flat and have only two dimensions (length and width), while 3D shapes have three dimensions (length, width, and depth/height) and occupy space. 2a) A sequence is an ordered list of numbers, called terms, that follow a specific pattern or rule. 2b) Sequences are important in real life because they help us to predict future values, model growth or decay, and understand patterns in various phenomena. 2c) Areas where sequences can be used include: • Finance (e.g., compound interest, loan payments) • Population growth modeling • Computer algorithms • Music patterns 2d) i) 4, 9, 16, 25 Step 1: Identify the pattern. The given terms are 2^2, 3^2, 4^2, 5^2. This is a sequence of perfect squares where the n-th term is T_n = (n+1)^2. Step 2: Calculate the next three terms. For n=5: T_5 = (5+1)^2 = 6^2 = 36 For n=6: T_6 = (6+1)^2 = 7^2 = 49 For n=7: T_7 = (7+1)^2 = 8^2 = 64 The next 3 terms are 36, 49, 64. ii) 3, 7, 11, 15 Step 1: Identify the pattern. The difference between consecutive terms is constant: 7-3=4, 11-7=4, 15-11=4. This is an arithmetic sequence with a first term a=3 and a common difference d=4. The n-th term is T_n = a + (n-1)d = 3 + (n-1)4 = 3 + 4n - 4 = 4n - 1. Step 2: Calculate the next three terms. For n=5: T_5 = 4(5) - 1 = 20 - 1 = 19 For n=6: T_6 = 4(6) - 1 = 24 - 1 = 23 For n=7: T_7 = 4(7) - 1 = 28 - 1 = 27 The next 3 terms are 19, 23, 27. 3a) Step 1: Substitute the value of x into the function. Given f(x) = x + 2, we need to find f(3). f(3) = 3 + 2 Step 2: Calculate the result. f(3) = 5 The value is 5. 3b) Step 1: Substitute the value of x into the function. Given f(x) = 9x, we need to find f(5). f(5) = 9 × 5 Step 2: Calculate the result. f(5) = 45 The value is 45. 3c) Step 1: Substitute the value of x into the function. Given f(x) = 9 - x, we need to find f(4). f(4) = 9 - 4 Step 2: Calculate the result. f(4) = 5 The value is 5. That's 2 down. 3 left today — send the next one.