answer:1. **Given Problem and Initial Setup:** We need to show that for ( -1 < x < 1 ), [ sum_{k=0}^{6} frac{1 - x^2}{1 - 2x cos left( frac{2pi k}{7} right) + x^2} = frac{7(1 + x^7)}{1 - x^7}. ] Then, we need to deduce that [ csc^2left( x + frac{pi}{7} right) + csc^2left(2x + frac{pi}{7} right) + csc^2left(3x + frac{pi}{7} right) = 8. ] 2. **Verification of the Given Sum:** Let's start by verifying the given sum. Consider the polynomial ( P(x) = prod_{k=0}^{6} (1 - 2x cos left( frac{2pi k}{7} right) + x^2) ). This polynomial has roots at the 7th roots of unity, excluding 1. Therefore, we can write: [ P(x) = frac{1 - x^7}{1 - x}. ] Differentiating ( P(x) ) with respect to ( x ) and evaluating at ( x = 0 ) gives us the sum of the reciprocals of the roots: [ sum_{k=0}^{6} frac{1}{1 - 2x cos left( frac{2pi k}{7} right) + x^2}. ] This sum can be simplified using the properties of roots of unity and symmetry. 3. **Using the Identity:** We use the identity: [ sum_{k=0}^{n-1} frac{1}{sin^2 left( x + frac{2kpi}{n} right)} = frac{n^2}{sin^2(nx)}. ] For ( n = 7 ), we have: [ sum_{k=0}^{6} frac{1}{sin^2 left( x + frac{2kpi}{7} right)} = frac{49}{sin^2(7x)}. ] 4. **Simplifying the Sum:** To find the sum in question, we take the limit as ( x to 0 ): [ frac{1}{2} lim_{x to 0} left( frac{49}{sin^2(7x)} - frac{1}{sin^2(x)} right). ] Using the small-angle approximation ( sin(x) approx x ) for ( x to 0 ), we get: [ frac{1}{2} left( frac{49}{(7x)^2} - frac{1}{x^2} right) = frac{1}{2} left( frac{49}{49x^2} - frac{1}{x^2} right) = frac{1}{2} left( frac{49 - 1}{x^2} right) = frac{1}{2} cdot frac{48}{x^2} = 8. ] 5. **Conclusion:** Therefore, we have shown that: [ csc^2 left( x + frac{pi}{7} right) + csc^2 left( 2x + frac{pi}{7} right) + csc^2 left( 3x + frac{pi}{7} right) = 8. ] (blacksquare) The final answer is ( boxed{ csc^2 left( x + frac{pi}{7} right) + csc^2 left( 2x + frac{pi}{7} right) + csc^2 left( 3x + frac{pi}{7} right) = 8 } ).

answer:To find out how long Will jogged for, we need to calculate the difference in calories before and after jogging, and then divide that by the number of calories burned per minute while jogging. Will's net calorie intake after jogging is 600 calories, and his initial calorie intake was 900 calories. The difference in calories is: 900 calories (before jogging) - 600 calories (after jogging) = 300 calories Now, we know that Will burns 10 calories per minute while jogging. To find out how long he jogged, we divide the total calories burned by the rate of calories burned per minute: 300 calories / 10 calories per minute = 30 minutes Therefore, Will jogged for boxed{30} minutes.

answer:To provide an answer to the question, we start by considering the definition of an irrational number. An irrational number is a number that cannot be expressed as a simple fraction. It's a number that has endless non-repeating digits to the right of the decimal point. One of the most famous and widely recognized irrational numbers is pi, which represents the ratio of the circumference of any circle to its diameter. Given this, we can write down an irrational number that is well-known and often used in mathematics: [ pi approx 3.14159... ] Therefore, the irrational number that I like and choose to write down is pi. So, the answer is encapsulated as: boxed{pi} (Note: The answer is not unique as there are infinitely many irrational numbers one could choose.)

answer:To analyze the given statements and derive a logical conclusion, let's break down the problem step by step. The statements are: 1. **Either the perpetrator left in a carriage, or the witness was mistaken.** 2. **If the perpetrator had an accomplice, then he left in a carriage.** 3. **The perpetrator either had no accomplice and no key, or had both an accomplice and a key.** 4. **The perpetrator had a key.** Step-by-Step Analysis: 1. **Translate Statements into Logical Expressions:** - Let ( P ) denote "the perpetrator left in a carriage". - Let ( W ) denote "the witness was mistaken". - Let ( A ) denote "the perpetrator had an accomplice". - Let ( K ) denote "the perpetrator had a key". 2. **Express the statements logically:** - Statement 1: ( P lor W ) - Statement 2: ( A Rightarrow P ) - Statement 3: ( neg A land neg K lor (A land K) ) - Statement 4: ( K ) 3. **Derive implications from Statement 3:** - ( neg A land neg K lor A land K ) states that either ( A ) and ( K ) are both false, or both are true. - Given ( K ) is true (Statement 4), ( A ) must also be true. Hence, ( A land K). 4. **Implication of ( A ):** - If ( A ) is true, by Statement 2 (( A Rightarrow P )), ( P ) must be true. 5. **The combined logical expressions:** - From Steps 1 to 4, we have: ( P ), ( A ), ( K ). - Statement 1 tells us ( P lor W). Since ( P ) is true, this is satisfied regardless of ( W ). 6. **Final conclusion:** - ( A ), ( K ), and ( P ) are true. - Thus, the accurate conclusion is ( P ) (the perpetrator left in a carriage). # Conclusion: [ boxed{text{The perpetrator left in a carriage.}} ]