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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Problem 7: Write down in its simplest form the complete expansion of (x-(1)/(2))^6. Hence, by taking x=(1)/(400) in this expansion, show that ((199)/(400))^6 = 0.01516, correct to five places of decimals. Step 1: Expand (x-(1)/(2))^6 using the binomial theorem. (x-(1)/(2))^6 = 60x^6(-(1)/(2))^0 + 61x^5(-(1)/(2))^1 + 62x^4(-(1)/(2))^2 + 63x^3(-(1)/(2))^3 + 64x^2(-(1)/(2))^4 + 65x^1(-(1)/(2))^5 + 66x^0(-(1)/(2))^6 = 1 · x^6 · 1 + 6 · x^5 · (-(1)/(2)) + 15 · x^4 · (1)/(4) + 20 · x^3 · (-(1)/(8)) + 15 · x^2 · (1)/(16) + 6 · x · (-(1)/(32)) + 1 · 1 · (1)/(64) = x^6 - 3x^5 + (15)/(4)x^4 - (5)/(2)x^3 + (15)/(16)x^2 - (3)/(16)x + (1)/(64) The complete expansion is x^6 - 3x^5 + (15)/(4)x^4 - (5)/(2)x^3 + (15)/(16)x^2 - (3)/(16)x + (1)/(64). Step 2: Substitute x=(1)/(400) into the expansion. When x=(1)/(400), the expression (x-(1)/(2))^6 becomes ((1)/(400)-(1)/(2))^6 = ((1)/(400)-(200)/(400))^6 = (-(199)/(400))^6 = ((199)/(400))^6. We evaluate the expansion with x=(1)/(400) = 0.0025. Since x is small, we can use the terms in ascending powers of x: (-(1)/(2)+x)^6 = (1)/(64) - (3)/(16)x + (15)/(16)x^2 - (5)/(2)x^3 + Substitute x=0.0025: (1)/(64) = 0.015625 -(3)/(16)x = -(3)/(16)(0.0025) = -0.00046875 (15)/(16)x^2 = (15)/(16)(0.0025)^2 = (15)/(16)(0.00000625) = 0.000005859375 The terms with x^3 and higher powers are very small and will not affect the fifth decimal place. Summing these values: 0.015625 - 0.00046875 + 0.000005859375 = 0.015162109375 Rounding to five decimal places, we get 0.01516. Thus, ((199)/(400))^6 ≈ 0.01516. Problem 8: The ratio of the third to the fourth term in the expansion of (2+3x)^n in ascending powers of x is 5:14 when x=(2)/(5). Find n. Step 1: Write the general term T_r+1 for (2+3x)^n. T_r+1 = nr (2)^n-r (3x)^r Step 2: Write the third term (T_3) and the fourth term (T_4). For T_3, r=2: T_3 = n2 (2)^n-2 (3x)^2 = n2 2^n-2 9x^2 For T_4, r=3: T_4 = n3 (2)^n-3 (3x)^3 = n3 2^n-3 27x^3 Step 3: Set up the ratio (T_3)/(T_4) and simplify. (T_3)/(T_4) = (n)/(2) 2^n