The first three terms of an arithmetic progression (AP) are p/6, 2p, 6 and p respectively. (a) Given that the common difference of the AP is not zero, find the value of p. (b) Find the 20th term of the AP. (c) Calculate the sum of the first 20 terms of the AP.

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Asked on June 4, 2026Mathematics

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.

Accepted Answer · 2026-06-04T17:31:31.717Z

a) Step 1: Identify the terms of the arithmetic progression (AP). The first three terms of the AP are given as T_1 = (p^2)/(6), T_2 = 2p-6, and T_3 = p. Step 2: Use the property of a common difference in an AP. In an AP, the common difference (d) is constant, meaning T_2 - T_1 = T_3 - T_2. Substitute the given terms into this equation: (2p-6) - (p^2)/(6) = p - (2p-6) Step 3: Simplify and solve the equation for p. 2p-6 - (p^2)/(6) = p - 2p + 6 2p-6 - (p^2)/(6) = -p + 6 Multiply the entire equation by 6 to eliminate the fraction: 6(2p-6) - p^2 = 6(-p+6) 12p - 36 - p^2 = -6p + 36 Rearrange the terms to form a quadratic equation: -p^2 + 12p + 6p - 36 - 36 = 0 -p^2 + 18p - 72 = 0 Multiply by -1 to make the leading coefficient positive: p^2 - 18p + 72 = 0 Step 4: Factor the quadratic equation. We need two numbers that multiply to 72 and add up to -18. These numbers are -6 and -12. (p-6)(p-12) = 0 This gives two possible values for p: p=6 or p=12 Step 5: Check the condition that the common difference is not zero. If p=6: T_1 = (6^2)/(6) = 6 T_2 = 2(6)-6 = 12-6 = 6 T_3 = 6 The terms are 6, 6, 6. The common difference d = 6-6 = 0. This contradicts the condition that the common difference is not zero. If p=12: T_1 = (12^2)/(6) = (144)/(6) = 24 T_2 = 2(12)-6 = 24-6 = 18 T_3 = 12 The terms are 24, 18, 12. The common difference d = 18-24 = -6. This is not zero. Therefore, the value of p is 12. p=12 b) Step 1: Determine the first term (a) and the common difference (d) using p=12. First term, a = T_1 = 24. Common difference, d = T_2 - T_1 = 18 - 24 = -6. Step 2: Use the formula for the n-th term of an AP, T_n = a + (n-1)d. We need to find the 20th term, so n=20. T_20 = a + (20-1)d T_20 = 24 + (19)(-6) T_20 = 24 - 114 T_20 = -90 T_20=-90 c) Step 1: Use the formula for the sum of the first n terms of an AP, S_n = (n)/(2)(a + T_n). We need to calculate the sum of the first 20 terms, so n=20. We have a=24 and T_20=-90. S_20 = (20)/(2)(24 + (-90)) S_20 = 10(24 - 90) S_20 = 10(-66) S_20 = -660 S_20=-660 That's all for today! Send me the next one 📸

Accepted Answer · 2026-06-04T17:31:31.717Z

a) Step 1: Identify the terms of the arithmetic progression (AP). The first three terms of the AP are given as T_1 = (p^2)/(6), T_2 = 2p-6, and T_3 = p. Step 2: Use the property of a common difference in an AP. In an AP, the common difference (d) is constant, meaning T_2 - T_1 = T_3 - T_2. Substitute the given terms into this equation: (2p-6) - (p^2)/(6) = p - (2p-6) Step 3: Simplify and solve the equation for p. 2p-6 - (p^2)/(6) = p - 2p + 6 2p-6 - (p^2)/(6) = -p + 6 Multiply the entire equation by 6 to eliminate the fraction: 6(2p-6) - p^2 = 6(-p+6) 12p - 36 - p^2 = -6p + 36 Rearrange the terms to form a quadratic equation: -p^2 + 12p + 6p - 36 - 36 = 0 -p^2 + 18p - 72 = 0 Multiply by -1 to make the leading coefficient positive: p^2 - 18p + 72 = 0 Step 4: Factor the quadratic equation. We need two numbers that multiply to 72 and add up to -18. These numbers are -6 and -12. (p-6)(p-12) = 0 This gives two possible values for p: p=6 or p=12 Step 5: Check the condition that the common difference is not zero. If p=6: T_1 = (6^2)/(6) = 6 T_2 = 2(6)-6 = 12-6 = 6 T_3 = 6 The terms are 6, 6, 6. The common difference d = 6-6 = 0. This contradicts the condition that the common difference is not zero. If p=12: T_1 = (12^2)/(6) = (144)/(6) = 24 T_2 = 2(12)-6 = 24-6 = 18 T_3 = 12 The terms are 24, 18, 12. The common difference d = 18-24 = -6. This is not zero. Therefore, the value of p is 12. p=12 b) Step 1: Determine the first term (a) and the common difference (d) using p=12. First term, a = T_1 = 24. Common difference, d = T_2 - T_1 = 18 - 24 = -6. Step 2: Use the formula for the n-th term of an AP, T_n = a + (n-1)d. We need to find the 20th term, so n=20. T_20 = a + (20-1)d T_20 = 24 + (19)(-6) T_20 = 24 - 114 T_20 = -90 T_20=-90 c) Step 1: Use the formula for the sum of the first n terms of an AP, S_n = (n)/(2)(a + T_n). We need to calculate the sum of the first 20 terms, so n=20. We have a=24 and T_20=-90. S_20 = (20)/(2)(24 + (-90)) S_20 = 10(24 - 90) S_20 = 10(-66) S_20 = -660 S_20=-660 That's all for today! Send me the next one 📸

The first three terms of an arithmetic progression (AP) are p/6, 2p, 6 and p respectively. (a) Given that the common difference of the AP is not zero, find the value of p. (b) Find the 20th term of the AP. (c) Calculate the sum of the first 20 terms of the AP.

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The first three terms of an arithmetic progression (AP) are p/6, 2p, 6 and p respectively. (a) Given that the common difference of the AP is not zero, find the value of p. (b) Find the 20th term of the AP. (c) Calculate the sum of the first 20 terms of the AP.

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