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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3.1.1 Write down the general term of this series. Step 1: Identify the first term (a) and the common ratio (r). The first term is a = -4. The common ratio is r = (2)/(-4) = -(1)/(2). Step 2: Write the general term using the formula T_n = ar^n-1. T_n = -4(-(1)/(2))^n-1 The general term is T_n = -4(-(1)/(2))^n-1. 3.1.2 Write the series in sigma notation. Step 1: Determine the number of terms (n) in the series. The last term is (1)/(32). Set T_n = (1)/(32): -4(-(1)/(2))^n-1 = (1)/(32) (-(1)/(2))^n-1 = (1)/(32 × (-4)) (-(1)/(2))^n-1 = -(1)/(128) Since 128 = 2^7, we can write: (-(1)/(2))^n-1 = (-(1)/(2))^7 n-1 = 7 n = 8 There are 8 terms in the series. Step 2: Write the series in sigma notation. _k=1^8 -4(-(1)/(2))^k-1 The series in sigma notation is _k=1^8 -4(-(1)/(2))^k-1. 3.1.3 Calculate the sum to infinity of this series. Step 1: Check if the sum to infinity exists. The common ratio is r = -(1)/(2). Since |r| = |-(1)/(2)| = (1)/(2) < 1, the sum to infinity exists. Step 2: Use the formula for the sum to infinity, S_ = (a)/(1-r). S_ = (-4)/(1 - (-1)2) S_ = (-4)/(1 + 1)2 S_ = (-4)/(3)2 S_ = -4 × (2)/(3) S_ = -(8)/(3) The sum to infinity is -(8)/(3). 3.2.1 Determine S_5. Given the formula for the sum of n terms: S_n = 32 - 32((1)/(2))^n. Substitute n=5 into the formula: S_5 = 32 - 32((1)/(2))^5 S_5 = 32 - 32((1)/(32)) S_5 = 32 - 1 S_5 = 31 The value of S_5 is 31. 3.2.2 How many terms must be added for the sum to be equal to (255)/(8)? Set S_n = (255)/(8): 32 - 32((1)/(2))^n = (255)/(8) 32 - (255)/(8) = 32((1)/(2))^n (256 - 255)/(8) = 32((1)/(2))^n (1)/(8) = 32((1)/(2))^n (1)/(8 × 32) = ((1)/(2))^n (1)/(256) = ((1)/(2))^n Since 256 = 2^8, we have: ((1)/(2))^8 = ((1)/(2))^n n = 8 The number of terms that must be added is 8. 3.2.3 If 2^n = p, determine the value of S_5-n - S_5+n in terms of p. Step 1: Rewrite S_n in terms of 2^n. S_n = 32 - 32((1)/(2))^n = 32 - (32)/(2^n) Given 2^n = p, so S_n = 32 - (32)/(p). Step 2: Determine S_5-n in terms of p. S_5-n = 32 - 32((1)/(2))^5-n S_5-n = 32 - 32 · (1)/(2^5-n) S_5-n = 32 - 32 · (1)/(2^5 · 2^-n) S_5-n = 32 - 32 · (2^n)/(32) S_5-n = 32 - 2^n Substitute 2^n = p: S_5-n = 32 - p Step 3: Determine S_5+n in terms of p. S_5+n = 32 - 32((1)/(2))^5+n S_5+n = 32 - 32 · (1)/(2^5+n) S_5+n = 32 - 32 · (1)/(2^5 · 2^n) S_5+n = 32 - 32 · (1)/(32 · 2^n) S_5+n = 32 - (1)/(2^n) Substitute 2^n = p: S_5+n = 32 - (1)/(p) Step 4: Calculate S_5-n - S_5+n. S_5-n - S_5+n = (32 - p) - (32 - (1)/(p)) S_5-n - S_5+n = 32 - p - 32 + (1)/(p) S_5-n - S_5+n = -p + (1)/(p) S_5-n - S_5+n = (1 - p^2)/(p) The value of S_5-n - S_5+n in terms of p is (1 - p^2)/(p). 3 done, 2 left today. You're making progress.