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 are the solutions for questions 2, 3, and 4. Question 2: a) If (^n C_2)/(^n P_3) = (1)/(10), find the value of n. Step 1: Write out the formulas for combinations and permutations. ^n C_r = (n!)/(r!(n-r)!) ^n P_r = (n!)/((n-r)!) Step 2: Apply the formulas to ^n C_2 and ^n P_3. ^n C_2 = (n!)/(2!(n-2)!) = (n(n-1)(n-2)!)/(2 · 1 · (n-2)!) = (n(n-1))/(2) ^n P_3 = (n!)/((n-3)!) = (n(n-1)(n-2)(n-3)!)/((n-3)!) = n(n-1)(n-2) Step 3: Substitute these expressions into the given equation. (n(n-1))/(2)n(n-1)(n-2) = (1)/(10) Step 4: Simplify the equation. Assuming n 0 and n 1 (which must be true for the terms to be defined, as n 3 for ^n P_3), we can cancel n(n-1) from the numerator and denominator. (1)/(2(n-2)) = (1)/(10) Step 5: Solve for n. 2(n-2) = 10 n-2 = (10)/(2) n-2 = 5 n = 5+2 n = 7 The value of n is: 7 b) Evaluate _p 2 - _p 9 + _p 4sqrt(3). Step 1: Use the logarithm properties a - b = (a)/(b) and a + b = (ab). _p 2 - _p 9 + _p 4sqrt(3) = _p ((2)/(9)) + _p 4sqrt(3) Step 2: Combine the terms. _p ((2)/(9) × 4sqrt(3)) _p (8sqrt(3)9) The evaluated expression is: _p (8sqrt(3)9) Question 3: The product of the 3rd and 7th terms of an arithmetic progression is 105. The 18th term divided by the 5th term is 5. The first term is positive. Find: a) second term b) Sum of the first 20 terms. Step 1: Define the terms of an arithmetic progression. Let the first term be a and the common difference be d. The n-th term is T_n = a + (n-1)d. T_3 = a+2d T_7 = a+6d T_5 = a+4d T_18 = a+17d Step 2: Set up equations based on the given information. 1) Product of 3rd and 7th terms is 105: (a+2d)(a+6d) = 105 2) 18th term divided by 5th term is 5: (a+17d)/(a+4d) = 5 Step 3: Solve equation (2) for a in terms of d or d in terms of a. a+17d = 5(a+4d) a+17d = 5a+20d 17d - 20d = 5a - a -3d = 4a d = -(4)/(3)a Step 4: Substitute the expression for d into equation (1). (a+2(-(4)/(3)a))(a+6(-(4)/(3)a)) = 105 (a-(8)/(3)a)(a-(24)/(3)a) = 105 ((3a-8a)/(3))((3a-24a)/(3)) = 105 (-(5a)/(3))(-7a) = 105 (35a^2)/(3) = 105 Step 5: Solve for a. 35a^2 = 105 × 3 35a^2 = 315 a^2 = (315)/(35) a^2 = 9 a = ± 3 Since the first term is positive, a=3. Step