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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You're on a roll — here's the solution for Question 3. Part 3.1: Consider the following linear number pattern: 23; 17; 11; p; -1; Step 1: Determine the common difference (d). For a linear number pattern, the common difference between consecutive terms is constant. d = T_2 - T_1 = 17 - 23 = -6 d = T_3 - T_2 = 11 - 17 = -6 The common difference is -6. 3.1.1: Write down the value of p Step 2: Calculate p. The term p is the term after 11. p = 11 + d p = 11 + (-6) p = 5 3.1.2: Determine the n^th term of the number pattern Step 3: Use the formula for the n^th term of a linear pattern. The formula is T_n = a + (n-1)d, where a is the first term and d is the common difference. Here, a = 23 and d = -6. T_n = 23 + (n-1)(-6) T_n = 23 - 6n + 6 T_n = -6n + 29 3.1.3: Determine the 15^th term of the sequence Step 4: Substitute n=15 into the n^th term formula. T_15 = -6(15) + 29 T_15 = -90 + 29 T_15 = -61 3.1.4: Which term will be equal to -121? Step 5: Set T_n = -121 and solve for n. -6n + 29 = -121 Subtract 29 from both sides: -6n = -121 - 29 -6n = -150 Divide by -6: n = (-150)/(-6) n = 25 The 25^th term will be equal to -121. Part 3.2: 2x+1, 18, 4x+5 are the first 3 terms of a linear number pattern 3.2.1: Determine the value of x Step 1: Use the property of a linear number pattern. In a linear pattern, the common difference between consecutive terms is constant. So, T_2 - T_1 = T_3 - T_2. 18 - (2x+1) = (4x+5) - 18 Step 2: Solve the equation for x. 18 - 2x - 1 = 4x + 5 - 18 17 - 2x = 4x - 13 Add 2x to both sides and add 13 to both sides: 17 + 13 = 4x + 2x 30 = 6x Divide by 6: x = (30)/(6) x = 5 Drop the next question!