answer:According to the given information, we have a_n = -1 + (n - 1) cdot frac{1}{5} = frac{n}{5} - frac{6}{5}. To find the first positive term, we need to solve for a_n > 0. This gives us n > 6. Therefore, the 7th term is the first positive term in the sequence. Hence, the answer is: boxed{D}. To solve this problem, we first use the formula for the nth term of an arithmetic sequence. Then, we set the nth term greater than 0 and solve for the value of n. This allows us to identify the first positive term in the sequence.

answer:Let F be the center of the square base, and P be the apex of the pyramid. Since it is a right pyramid, triangle PFB is a right triangle. 1. **Calculate the side of the square base**: The perimeter is 40 inches, so each side of the square is frac{40}{4} = 10 inches. 2. **Determine FB**: F is the center of the base, so FB is half the diagonal of the square. The diagonal of a square with side length s is ssqrt{2}; hence, FB = frac{10sqrt{2}}{2} = 5sqrt{2} inches. 3. **Apply Pythagorean theorem in triangle PFB**: [ PF = sqrt{PB^2 - FB^2} = sqrt{100 - 50} = sqrt{50} = 5sqrt{2} text{ inches}. ] Thus, the height of the pyramid from its peak to the center of its square base is boxed{5sqrt{2} text{ inches}}.

answer:To calculate the given expression sqrt{12}+2tan45°-sin60°-{(frac{1}{2})}^{-1}, we break it down step by step: 1. Simplify sqrt{12}: [ sqrt{12} = sqrt{4 cdot 3} = sqrt{4}sqrt{3} = 2sqrt{3} ] 2. Evaluate 2tan45°: [ 2tan45° = 2 cdot 1 = 2 ] 3. Calculate -sin60°: [ -sin60° = -frac{sqrt{3}}{2} ] 4. Simplify {(frac{1}{2})}^{-1}: [ {(frac{1}{2})}^{-1} = 2 ] Putting it all together: [ sqrt{12}+2tan45°-sin60°-{(frac{1}{2})}^{-1} = 2sqrt{3} + 2 - frac{sqrt{3}}{2} - 2 ] Combine like terms: [ = 2sqrt{3} - frac{sqrt{3}}{2} ] To combine the terms, find a common denominator, which is 2: [ = frac{4sqrt{3}}{2} - frac{sqrt{3}}{2} = frac{3sqrt{3}}{2} ] Therefore, the final answer is: [ boxed{frac{3sqrt{3}}{2}} ]

answer:1. Given the polynomial equation (x^4 - x - 504 = 0), we know it has four roots (r_1, r_2, r_3, r_4). 2. For each root (r_i), we have (r_i^4 = r_i + 504). This follows directly from substituting (r_i) into the polynomial equation. 3. We need to find the value of (S_4 = r_1^4 + r_2^4 + r_3^4 + r_4^4). 4. Using the equation (r_i^4 = r_i + 504), we can express (S_4) as: [ S_4 = r_1^4 + r_2^4 + r_3^4 + r_4^4 = (r_1 + 504) + (r_2 + 504) + (r_3 + 504) + (r_4 + 504) ] 5. Simplifying the above expression, we get: [ S_4 = (r_1 + r_2 + r_3 + r_4) + 4 cdot 504 ] 6. By Vieta's formulas, the sum of the roots of the polynomial (x^4 - x - 504 = 0) is zero (since the coefficient of (x^3) is zero). Therefore: [ r_1 + r_2 + r_3 + r_4 = 0 ] 7. Substituting this into the expression for (S_4), we get: [ S_4 = 0 + 4 cdot 504 = 2016 ] The final answer is (boxed{2016}).