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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You're on a roll — Question 7: Given a Geometric Progression (GP): (128)/(125), ..., 100, 250, 625. a) Find the number of terms. Step 1: Identify the first term and common ratio. The first term is a = (128)/(125). From the given terms 100, 250, 625, we can find the common ratio r. r = (250)/(100) = (5)/(2) We can verify this with the next pair: r = (625)/(250) = (5)/(2) Step 2: Use the formula for the n-th term of a GP. The last term is a_n = 625. The formula for the n-th term is a_n = ar^n-1. Substitute the values of a, r, and a_n: (128)/(125) ((5)/(2))^n-1 = 625 Step 3: Solve for n. Express the numbers as powers of their prime factors: 128 = 2^7, 125 = 5^3, 625 = 5^4. (2^7)/(5^3) ((5)/(2))^n-1 = 5^4 (2^7)/(5^3) 5^n-12^n-1 = 5^4 2^7-(n-1) · 5^(n-1)-3 = 5^4 2^8-n · 5^n-4 = 5^4 For this equality to hold, the powers of the bases must match. Equating the powers of 2: 8-n = 0 n = 8 Equating the powers of 5: n-4 = 4 n = 8 Both equations yield n=8. The number of terms is 8. b) Write down the first three terms. Step 1: Identify the first term. The first term is a_1 = a = (128)/(125). Step 2: Calculate the second term. The second term is a_2 = ar. a_2 = (128)/(125) × (5)/(2) = (64 × 2)/(25 × 5) × (5)/(2) = (64)/(25) Step 3: Calculate the third term. The third term is a_3 = ar^2. a_3 = (128)/(125) × ((5)/(2))^2 = (128)/(125) × (25)/(4) = (32 × 4)/(5 × 25) × (25)/(4) = (32)/(5) The first three terms are (128)/(125), (64)/(25), (32)/(5). Question 8: Which term of the sequence: 10(1)/(8), 6(3)/(4), 4(1)/(2), ... is (256)/(2187)? Step 1: Convert mixed numbers to improper fractions and identify the first term. 10(1)/(8) = (8 × 10 + 1)/(8) = (81)/(8) 6(3)/(4) = (4 × 6 + 3)/(4) = (27)/(4) 4(1)/(2) = (2 × 4 + 1)/(2) = (9)/(2) The sequence is (81)/(8), (27)/(4), (9)/(2), ... The first term is a = (81)/(8). Step 2: Find the common ratio r. r = second termfirst term = (27/4)/(81/8) = (27)/(4) × (8)/(81) = (27)/(81) × (8)/(4) = (1)/(3) × 2 = (2)/(3) Verify with the next pair: r = third termsecond term = (9/2)/(27/4) = (9)/(2) × (4)/(27) = (9)/(27) × (4)/(2) = (1)/(3) × 2 = (2)/(3) The common ratio is r = (2)/(3). Step 3: Use the formula for the n-th term of a GP and solve for n. We want to find n such that a_n = (256)/(2187). The formula is a_n = ar^n-1. (81)/(8) ((2)/(3))^n-1 = (256)/(2187) Step 4: Express numbers as powers of their prime factors and solve for n. 81 = 3^4 8 = 2^3 256 = 2^8 2187 = 3^7 Substitute these into the equation: (3^4)/(2^3) ((2)/(3))^n-1 = (2^8)/(3^7) (3^4)/(2^3) 2^n-13^n-1 = (2^8)/(3^7) 2^(n-1)-3 · 3^4-(n-1) = 2^8 · 3^-7 2^n-4 · 3^5-n = 2^8 · 3^-7 Equating the powers of 2: n-4 = 8 n = 12 Equating the powers of 3: 5-n = -7 n = 5+7 n = 12 Both equations give n=12. The term is the 12th term. Drop the next question.