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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Let's determine which of these sequences are Arithmetic Progressions (APs). An AP has a constant difference between consecutive terms. (i) 2, 4, 8, 16, Step 1: Calculate the differences between consecutive terms. a_2 - a_1 = 4 - 2 = 2 a_3 - a_2 = 8 - 4 = 4 Since the differences are not constant (2 ≠ 4), this sequence is not an AP. (ii) 2, (5)/(2), 3, (7)/(2), Step 1: Calculate the differences between consecutive terms. a_2 - a_1 = (5)/(2) - 2 = (5)/(2) - (4)/(2) = (1)/(2) a_3 - a_2 = 3 - (5)/(2) = (6)/(2) - (5)/(2) = (1)/(2) a_4 - a_3 = (7)/(2) - 3 = (7)/(2) - (6)/(2) = (1)/(2) Step 2: Determine if it's an AP and find the common difference. The common difference is constant, d = (1)/(2). So, this is an AP. Step 3: Write the next three terms. a_5 = (7)/(2) + (1)/(2) = (8)/(2) = 4 a_6 = 4 + (1)/(2) = (9)/(2) a_7 = (9)/(2) + (1)/(2) = (10)/(2) = 5 This is an AP with d = (1)/(2). The next three terms are 4, (9)/(2), 5. (iii) -1.2, -3.2, -5.2, -7.2, Step 1: Calculate the differences between consecutive terms. a_2 - a_1 = -3.2 - (-1.2) = -3.2 + 1.2 = -2.0 a_3 - a_2 = -5.2 - (-3.2) = -5.2 + 3.2 = -2.0 a_4 - a_3 = -7.2 - (-5.2) = -7.2 + 5.2 = -2.0 Step 2: Determine if it's an AP and find the common difference. The common difference is constant, d = -2.0. So, this is an AP. Step 3: Write the next three terms. a_5 = -7.2 + (-2.0) = -9.2 a_6 = -9.2 + (-2.0) = -11.2 a_7 = -11.2 + (-2.0) = -13.2 This is an AP with d = -2.0. The next three terms are -9.2, -11.2, -13.2. (iv) -10, -6, -2, 2, Step 1: Calculate the differences between consecutive terms. a_2 - a_1 = -6 - (-10) = -6 + 10 = 4 a_3 - a_2 = -2 - (-6) = -2 + 6 = 4 a_4 - a_3 = 2 - (-2) = 2 + 2 = 4 Step 2: Determine if it's an AP and find the common difference. The common difference is constant, d = 4. So, this is an AP. Step 3: Write the next three terms. a_5 = 2 + 4 = 6 a_6 = 6 + 4 = 10 a_7 = 10 + 4 = 14 This is an AP with d = 4. The next three terms are 6, 10, 14. (v) 3, 3+sqrt(2), 3+2sqrt(2), 3+3sqrt(2), Step 1: Calculate the differences between consecutive terms. a_2 - a_1 = (3+sqrt(2)) - 3 = sqrt(2) a_3 - a_2 = (3+2sqrt(2)) - (3+sqrt(2)) = 3+2sqrt(2) - 3 - sqrt(2) = sqrt(2) a_4 - a_3 = (3+3sqrt(2)) - (3+2sqrt(2)) = 3+3sqrt(2) - 3 - 2sqrt(2) = sqrt(2) Step 2: Determine if it's an AP and find the common difference. The common difference is constant, d = sqrt(2). So, this is an AP. Step 3: Write the next three terms. a_5 = (3+3sqrt(2)) + sqrt(2) = 3+4sqrt(2) a_6 = (3+4sqrt(2)) + sqrt(2) = 3+5sqrt(2) a_7 = (3+5sqrt(2)) + sqrt(2) = 3+6sqrt(2) This is an AP with d = sqrt(2). The next three terms are 3+4sqrt(2), 3+5sqrt(2), 3+6sqrt(2). (vi) 0.2, 0.22, 0.222, 0.2222, Step 1: Calculate the differences between consecutive terms. a_2 - a_1 = 0.22 - 0.2 = 0.02 a_3 - a_2 = 0.222 - 0.22 = 0.002 Since the differences are not constant (0.02 ≠ 0.002), this sequence is not an AP. (vii) 0, -4, -8, -12, Step 1: Calculate the differences between consecutive terms. a_2 - a_1 = -4 - 0 = -4 a_3 - a_2 = -8 - (-4) = -8 + 4 = -4 a_4 - a_3 = -12 - (-8) = -12 + 8 = -4 Step 2: Determine if it's an AP and find the common difference. The common difference is constant, d = -4. So, this is an AP. Step 3: Write the next three terms. a_5 = -12 + (-4) = -16 a_6 = -16 + (-4) = -20 a_7 = -20 + (-4) = -24 This is an AP with d = -4. The next three terms are -16, -20, -24. (viii) -(1)/(2), -(1)/(2), -(1)/(2), -(1)/(2), Step 1: Calculate the differences between consecutive terms. a_2 - a_1 = -(1)/(2) - (-(1)/(2)) = 0 a_3 - a_2 = -(1)/(2) - (-(1)/(2)) = 0 a_4 - a_3 = -(1)/(2) - (-(1)/(2)) = 0 Step 2: Determine if it's an AP and find the common difference. The common difference is constant, d = 0. So, this is an AP. Step 3: Write the next three terms. a_5 = -(1)/(2) + 0 = -(1)/(2) a_6 = -(1)/(2) + 0 = -(1)/(2) a_7 = -(1)/(2) + 0 = -(1)/(2) This is an AP with d = 0. The next three terms are -(1)/(2), -(1)/(2), -(1)/(2). Last free one today — make it count tomorrow, or type /upgrade for unlimited.