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
i) Step 1: Write down the binomial expansion formula. The binomial expansion for (1+y)^n is given by: (1+y)^n = 1 + ny + (n(n-1))/(2!)y^2 + In this problem, y = (1)/(3)x and n=8. Step 2: Expand the given expression up to the x^2 term. Substitute y = (1)/(3)x and n=8 into the formula: (1+(1)/(3)x)^8 = 1 + 8((1)/(3)x) + (8(8-1))/(2!)((1)/(3)x)^2 + (1+(1)/(3)x)^8 = 1 + (8)/(3)x + (8 × 7)/(2)((1)/(9)x^2) + (1+(1)/(3)x)^8 = 1 + (8)/(3)x + 28((1)/(9)x^2) + (1+(1)/(3)x)^8 = 1 + (8)/(3)x + (28)/(9)x^2 + Step 3: Compare the expanded form with the given expression to find a and b. Given: (1+(1)/(3)x)^8 = 1 + (2)/(3)ax - (4)/(9)bx^2 + Comparing the coefficient of x: (8)/(3)x = (2)/(3)ax 8 = 2a a = (8)/(2) a = 4 Comparing the coefficient of x^2: (28)/(9)x^2 = -(4)/(9)bx^2 28 = -4b b = (28)/(-4) b = -7 The values of the constants are a=4, b=-7. ii) Step 1: Understand the problem. We are given the sum of the first n terms of a sequence, S_n = (n)/(2)(3n+1). We need to find the sum of the terms from the 10^th term to the 30^th term. This can be calculated as S_30 - S_9. Step 2: Calculate S_30. Substitute n=30 into the formula for S_n: S_30 = (30)/(2)(3(30)+1) S_30 = 15(90+1) S_30 = 15(91) S_30 = 1365 Step 3: Calculate S_9. Substitute n=9 into the formula for S_n: S_9 = (9)/(2)(3(9)+1) S_9 = (9)/(2)(27+1) S_9 = (9)/(2)(28) S_9 = 9 × 14 S_9 = 126 Step 4: Calculate the sum from the 10^th term to the 30^th term. Sum = S_30 - S_9 Sum = 1365 - 126 Sum = 1239 The sum of the terms from the 10^th term to the 30^th term is 1239.