answer:1. **Define the Lucas and Fibonacci sequences:** - Lucas numbers: ( L_0 = 2, L_1 = 1, L_2 = 3, L_3 = 4, L_4 = 7, ldots ) - Fibonacci numbers: ( F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, ldots ) 2. **Identify the problem:** - We need to find the probability that the sequence starting from a randomly chosen integer ( k in [0, K] ) contains 4, 5, or 6. - We need to express this probability in the form ( frac{a - b sqrt{c}}{d} ) and find ( a + b + c + d ). 3. **Count the number of good integers:** - An integer ( k ) is good if the sequence starting from ( k ) contains 4, 5, or 6. - We claim that the number of good integers in the range ([0, L_n)) is ( 3F_{n-3} ). 4. **Base cases:** - For ( L_3 = 4 ) and ( L_4 = 7 ), it is straightforward to verify the claim. 5. **Inductive step:** - Assume the claim holds for ( L_n ) and ( L_{n-1} ). - For any integer ( k in [L_n, L_{n+1}) ), ( k ) is good if ( k - L_n ) is good. - Thus, the number of good integers in ([L_n, L_{n+1})) is the number of good integers in ([0, L_{n-1})), which is ( 3F_{n-4} ). - Therefore, the number of good integers less than ( L_{n+1} ) is ( 3F_{n-3} + 3F_{n-4} = 3F_{n-2} ). 6. **Proportion of good integers:** - For large ( K ), let ( L_{a_1} ) be the greatest Lucas number less than or equal to ( K ). - There are ( 3F_{a_1-3} ) good integers less than ( L_{a_1} ). - Let ( L_{a_2} ) be the greatest Lucas number less than or equal to ( K - L_{a_1} ). - There are ( 3F_{a_2-3} ) good integers less than ( L_{a_2} ), so ( 3F_{a_2-3} ) good integers in ([L_{a_1}, L_{a_1} + L_{a_2})). 7. **Asymptotic behavior:** - The proportion of good integers approaches the ratio between ( L_n ) and ( 3F_{n-3} ) for large ( n ). - Using the closed forms: [ L_n approx left( frac{1 + sqrt{5}}{2} right)^n + left( frac{1 - sqrt{5}}{2} right)^n ] [ F_n approx frac{1}{sqrt{5}} left( left( frac{1 + sqrt{5}}{2} right)^n - left( frac{1 - sqrt{5}}{2} right)^n right) ] - The ratio is: [ frac{3F_{n-3}}{L_n} approx frac{3 cdot frac{1}{sqrt{5}} left( left( frac{1 + sqrt{5}}{2} right)^{n-3} - left( frac{1 - sqrt{5}}{2} right)^{n-3} right)}{left( frac{1 + sqrt{5}}{2} right)^n} ] [ approx frac{3 cdot frac{1}{sqrt{5}} left( frac{1 + sqrt{5}}{2} right)^{n-3}}{left( frac{1 + sqrt{5}}{2} right)^n} = frac{3}{sqrt{5}} cdot left( frac{1}{left( frac{1 + sqrt{5}}{2} right)^3} right) ] [ = frac{3}{sqrt{5}} cdot frac{2^3}{(1 + sqrt{5})^3} = frac{3 cdot 8}{sqrt{5} cdot (1 + sqrt{5})^3} ] [ = frac{24}{sqrt{5} cdot (1 + sqrt{5})^3} ] 8. **Simplify the expression:** - Using the binomial expansion and simplification: [ (1 + sqrt{5})^3 = 1 + 3sqrt{5} + 3 cdot 5 + sqrt{5}^3 = 1 + 3sqrt{5} + 15 + 5sqrt{5} = 16 + 8sqrt{5} ] [ frac{24}{sqrt{5} cdot (16 + 8sqrt{5})} = frac{24}{sqrt{5} cdot 8(2 + sqrt{5})} = frac{3}{sqrt{5} cdot (2 + sqrt{5})} ] [ = frac{3}{2sqrt{5} + 5} cdot frac{2sqrt{5} - 5}{2sqrt{5} - 5} = frac{3(2sqrt{5} - 5)}{(2sqrt{5})^2 - 5^2} = frac{3(2sqrt{5} - 5)}{20 - 25} = frac{3(2sqrt{5} - 5)}{-5} ] [ = frac{3(2sqrt{5} - 5)}{-5} = frac{6sqrt{5} - 15}{-5} = -frac{6sqrt{5}}{5} + 3 = frac{15 - 6sqrt{5}}{5} ] 9. **Identify ( a, b, c, d ):** - ( a = 15 ) - ( b = 6 ) - ( c = 5 ) - ( d = 5 ) 10. **Sum ( a + b + c + d ):** [ a + b + c + d = 15 + 6 + 5 + 5 = 31 ] The final answer is ( boxed{31} ).

answer:I'm sorry for any confusion, but as a cooking recipe AI, I specialize in providing cooking and food-related advice. I can guide you through preparing a Thanksgiving dinner, suggest recipes, or offer tips on cooking techniques. However, I'm not equipped to provide book recommendations on leadership or other non-cooking related topics.

answer:Absolutely, I'd be happy to suggest a few cookbooks that can help you improve your cooking skills: 1. The Joy of Cooking by Irma S. Rombauer - This is a classic cookbook that covers a wide range of recipes and cooking techniques. It's a great reference for any home cook. 2. How to Cook Everything by Mark Bittman - This book is another great all-around guide. It covers everything from basic techniques to more complex recipes. 3. Salt, Fat, Acid, Heat by Samin Nosrat - This book teaches the fundamentals of cooking and how to use these four elements to make your food taste better. 4. Mastering the Art of French Cooking by Julia Child - This is a classic cookbook that focuses on French cuisine. It's a bit more advanced, but it's excellent for learning new techniques and recipes. 5. The Professional Chef by the Culinary Institute of America - This is a comprehensive guide that covers a wide range of techniques and recipes. It's used by culinary students and is a great resource for anyone looking to improve their skills. Remember, improving cooking skills takes practice. Don't be afraid to make mistakes and experiment with different recipes and techniques. Happy cooking!

answer:Absolutely, baking is a wonderful skill to develop. Here are some excellent cookbooks that focus on baking: 1. The Baking Bible by Rose Levy Beranbaum - This book covers a wide range of baking techniques, from bread to pastries and everything in between. 2. Flour: Spectacular Recipes from Boston's Flour Bakery + Cafe by Joanne Chang - This book offers a collection of approachable and high-quality recipes from the famed Boston bakery. 3. Baking: From My Home to Yours by Dorie Greenspan - Dorie Greenspan is a well-respected baker, and this book provides home cooks with hundreds of simple, straightforward recipes. 4. The Bread Baker's Apprentice by Peter Reinhart - If you're interested in bread baking, this book is a must-read. It offers a variety of bread recipes and detailed instructions. 5. BraveTart: Iconic American Desserts by Stella Parks - This book is perfect if you're interested in classic American desserts. It also provides a deep dive into the history of these desserts, making it a fun read as well. Remember, baking is a science, so follow the recipes closely until you're comfortable enough to start experimenting. Enjoy your baking journey!