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
a) (i) Find the first dose and the common difference. Let the first dose be a_1 and the common difference be d. For an arithmetic progression, the n-th term is given by a_n = a_1 + (n-1)d. Given: Dose on Day 3 (a_3) is 15 mg. a_3 = a_1 + (3-1)d a_1 + 2d = 15 (1) Dose on Day 7 (a_7) is 27 mg. a_7 = a_1 + (7-1)d a_1 + 6d = 27 (2) Subtract equation (1) from equation (2): (a_1 + 6d) - (a_1 + 2d) = 27 - 15 4d = 12 d = (12)/(4) d = 3 mg Substitute d=3 into equation (1): a_1 + 2(3) = 15 a_1 + 6 = 15 a_1 = 15 - 6 a_1 = 9 mg The first dose is 9 mg and the common difference is 3 mg. (ii) What will be the dose on Day 12? Using the formula a_n = a_1 + (n-1)d with a_1 = 9 mg, d = 3 mg, and n=12: a_12 = 9 + (12-1)3 a_12 = 9 + (11)3 a_12 = 9 + 33 a_12 = 42 mg The dose on Day 12 will be 42 mg. (iii) Calculate the total medication administered in the first 2 weeks (14 days). The sum of the first n terms of an arithmetic progression is given by S_n = (n)/(2)(2a_1 + (n-1)d). Here, n=14, a_1 = 9 mg, and d = 3 mg. S_14 = (14)/(2)(2(9) + (14-1)3) S_14 = 7(18 + (13)3) S_14 = 7(18 + 39) S_14 = 7(57) S_14 = 399 mg The total medication administered in the first 2 weeks is 399 mg. b) Find the first four terms in the expansion of (1 - 2x)^n in ascending power of x. Using the binomial expansion formula (a+b)^n = n0a^n b^0 + n1a^n-1b^1 + n2a^n-2b^2 + n3a^n-3b^3 + Here, a=1 and b=-2x. First term (k=0): n0(1)^n(-2x)^0 = 1 · 1 · 1 = 1 Second term (k=1): n1(1)^n-1(-2x)^1 = n · 1 · (-2x) = -2nx Third term (k=2): n2(1)^n-2(-2x)^2 = (n(n-1))/(2!) · 1 · (4x^2) = (n(n-1))/(2) · 4x^2 = 2n(n-1)x^2 Fourth term (k=3): n3(1)^n-3(-2x)^3 = (n(n-1)(n-2))/(3!) · 1 · (-8x^3) = (n(n-1)(n-2))/(6) · (-8x^3) = -(4n(n-1)(n-2))/(3)x^3 The first four terms are: 1 - 2nx + 2n(n-1)x^2 - (4n(n-1)(n-2))/(3)x^3 3 done, 2 left today. You're making progress.