answer:1. Let ( n ) be the common difference of the arithmetic progression. Since ( 1, b, c ) is an arithmetic progression, we have: [ b = 1 + n ] [ c = 1 + 2n ] 2. Since ( 1, c, b ) forms a geometric progression, the ratio between consecutive terms must be the same. Therefore: [ frac{c}{1} = frac{b}{c} ] Substituting ( b ) and ( c ) from the arithmetic progression: [ frac{1 + 2n}{1} = frac{1 + n}{1 + 2n} ] 3. Cross-multiplying to solve for ( n ): [ (1 + 2n)^2 = 1 + n ] Expanding and simplifying: [ 1 + 4n + 4n^2 = 1 + n ] [ 4n^2 + 4n = n ] [ 4n^2 + 3n = 0 ] Factoring out ( n ): [ n(4n + 3) = 0 ] 4. Solving for ( n ): [ n = 0 quad text{or} quad 4n + 3 = 0 ] [ n = 0 quad text{or} quad n = -frac{3}{4} ] 5. Since ( b ) and ( c ) are not both equal to 1, ( n neq 0 ). Therefore: [ n = -frac{3}{4} ] 6. Substituting ( n ) back into the expressions for ( b ) and ( c ): [ b = 1 + n = 1 - frac{3}{4} = frac{1}{4} ] [ c = 1 + 2n = 1 + 2left(-frac{3}{4}right) = 1 - frac{3}{2} = -frac{1}{2} ] 7. Calculating ( 100(b - c) ): [ 100(b - c) = 100left(frac{1}{4} - left(-frac{1}{2}right)right) = 100left(frac{1}{4} + frac{1}{2}right) = 100left(frac{1}{4} + frac{2}{4}right) = 100left(frac{3}{4}right) = 75 ] The final answer is ( boxed{75} )

answer:1. **Calculate the current average score**: Leon's current test scores are 72, 68, 75, 81, and 79. The average of these scores is calculated as: [ text{Average} = frac{72 + 68 + 75 + 81 + 79}{5} = frac{375}{5} = 75 ] 2. **Determine the desired average score**: Leon wants to raise his average by 5 points. Therefore, the target average score after the next test is: [ 75 + 5 = 80 ] 3. **Calculate the total score required to achieve the desired average**: After the next test, Leon will have taken 6 tests. To achieve an average of 80, the total score for all 6 tests must be: [ 80 times 6 = 480 ] 4. **Determine the score needed on the next test**: The sum of his current scores is 375. To find out the score needed on his next test to reach a total of 480, we calculate: [ text{Required score} = 480 - 375 = 105 ] Thus, the minimum score Leon needs on his next test to achieve his goal is 105. Conclusion: Leon needs to score at least 105 on his next test to raise his average score by 5 points, given his previous performance. The final answer is boxed{textbf{(C)} 105}

answer:In triangle (PQR), since (PQ = PR = 17) cm and (QR = 16) cm, triangle (PQR) is isosceles. The altitude (PS) from vertex (P) to the base (QR) not only bisects (QR) but is also perpendicular to (QR). Let (S) be the midpoint of (QR). Since (QR = 16) cm, the half-length of (QR) is: [ QS = frac{QR}{2} = frac{16}{2} = 8 text{ cm} ] We now use Pythagoras' theorem in triangle (PQS) to find (PS) (the height of (PQR)). Since (PQ = 17) cm and (QS = 8) cm, [ PS = sqrt{PQ^2 - QS^2} = sqrt{17^2 - 8^2} = sqrt{289 - 64} = sqrt{225} = 15 text{ cm} ] Finally, compute the area of (PQR) using the formula for the area of a triangle: [ text{Area} = frac{1}{2} times text{Base} times text{Height} = frac{1}{2} times 16 times 15 = 120 text{ square centimeters} ] Thus, the area of triangle (PQR) is (boxed{120} ) square centimeters.

answer:Let the roots of the polynomial be -p, -q, -r, and -s where p, q, r, s are positive integers. Therefore, we can write: [ f(x) = (x + p)(x + q)(x + r)(x + s) ] We know that f(1) = (1 + p)(1 + q)(1 + r)(1 + s) = 1 + a + b + c + d = 2024. After factorizing 2024, we find that: [ 2024 = 2^3 times 11 times 23 ] Thus, we can set the expressions (1+p), (1+q), (1+r), (1+s) to be 8, 11, 23, 12 respectively, leading to p=7, q=10, r=22, s=11 in some order. Therefore, [ f(x) = (x+7)(x+10)(x+22)(x+11) ] Now, we find d by calculating the product of the roots with their signs changed: [ d = 7 cdot 10 cdot 22 cdot 11 = 17020 ] Thus, d = boxed{17020}.