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question:Given the geometric sequence {a<sub>n</sub>}, where a<sub>2</sub> = 2 and a<sub>5</sub> = frac{1}{4}, find the value of a<sub>1</sub>a<sub>2</sub> + a<sub>2</sub>a<sub>3</sub> + ... + a<sub>5</sub>a<sub>6</sub>.
answer:Let q be the common ratio of the geometric sequence {a<sub>n</sub>}. Since a<sub>2</sub> = 2 and a<sub>5</sub> = frac{1}{4}, frac{1}{4} = 2 times q^3 Solving for q, we get q = frac{1}{2}. Thus, a<sub>1</sub> = frac{a_2}{q} = frac{2}{frac{1}{2}} = 4. Then, a<sub>1</sub>a<sub>2</sub> = 8. For n ≥ 2, frac{a_n a_{n+1}}{a_{n-1} a_n} = q^2 = frac{1}{4} Hence, a<sub>1</sub>a<sub>2</sub> + a<sub>2</sub>a<sub>3</sub> + ... + a<sub>5</sub>a<sub>6</sub> = frac{8 times (1 - frac{1}{4^5})}{1 - frac{1}{4}} = frac{341}{32}. Therefore, the answer is boxed{frac{341}{32}}. This problem tests the understanding of the general term formula, sum formula, and properties of geometric sequences. It also assesses reasoning and computational abilities, making it a moderately difficult question.
question:Given that S_{n} is the sum of the first n terms of the arithmetic sequence {a_{n}}, if a_{2}+a_{5}+a_{8}=12, then S_{9} is equal to ( ) A: 18 B: 36 C: 72 D: Cannot be determined
answer:By the properties of an arithmetic sequence, we have a_{2}+a_{5}+a_{8}=3a_{5}=12. Solving for a_{5}, we get a_{5}=4. Using the sum formula, we have S_{9}= frac {9(a_{1}+a_{9})}{2}. Since a_{5} is the middle term of a_{1},a_{2},...,a_{9}, we have a_{1}+a_{9}=2a_{5}. Thus, S_{9}= frac {9(2a_{5})}{2}=9a_{5}=9 times 4 = boxed{36}.
question:What is the molecular weight of some moles of C6H8O7 if the molecular weight of 7 moles is 1344?
answer:To find the molecular weight of one mole of C6H8O7, we need to divide the given molecular weight of 7 moles by 7. Molecular weight of 7 moles of C6H8O7 = 1344 g/mol Molecular weight of 1 mole of C6H8O7 = 1344 g/mol ÷ 7 Molecular weight of 1 mole of C6H8O7 = 192 g/mol Therefore, the molecular weight of one mole of C6H8O7 is boxed{192} g/mol. If you want to find the molecular weight of a different number of moles, you would simply multiply this value by the number of moles you're interested in.
question:The perimeter of a rectangle is 46 centimeters (cm), and its width is 7 centimeters (cm) longer than its length. What is the width of this rectangle?
answer:Let's denote the length of the rectangle as L cm and the width as W cm. According to the problem, the width is 7 cm longer than the length, so we can write: W = L + 7 The perimeter of a rectangle is given by the formula: Perimeter = 2 * (Length + Width) We are given that the perimeter is 46 cm, so we can write: 46 = 2 * (L + W) Now we can substitute the expression for W into this equation: 46 = 2 * (L + (L + 7)) Simplify the equation: 46 = 2 * (2L + 7) Divide both sides by 2 to solve for L: 23 = 2L + 7 Subtract 7 from both sides: 16 = 2L Divide both sides by 2: L = 8 Now that we have the length, we can find the width using the relationship W = L + 7: W = 8 + 7 W = 15 So the width of the rectangle is boxed{15} cm.