answer:First, we calculate the initial terms of the sequence starting from 2017: - The sum of the cubes of the digits of 2017 is 2^3 + 0^3 + 1^3 + 7^3 = 8 + 0 + 1 + 343 = 352. - Next term, sum of the cubes of the digits of 352 is 3^3 + 5^3 + 2^3 = 27 + 125 + 8 = 160. - Next term, sum of the cubes of the digits of 160 is 1^3 + 6^3 + 0^3 = 1 + 216 + 0 = 217. - Next term, sum of the cubes of the digits of 217 is 2^3 + 1^3 + 7^3 = 8 + 1 + 343 = 352. We see that the sequence repeats from 352 again. Hence, the sequence becomes periodic with a period of 3 after the very first term. Given 2017 as the starting point, the actual cycle starts from the second term: - The 2017th term corresponds to the second term in the cycle due to 2017 equiv 2 pmod{3}, after counting the first term as not part of the repeating cycle. Therefore, the 2017th term is equal to the first term of the cycle, which is boxed{352}.

answer:To find the speed of the train, we can use the formula: Speed = Distance / Time The distance the train covers to pass the electric pole is equal to the length of the train, which is 500 meters. The time taken to cross the electric pole is given as 20 seconds. Plugging in the values, we get: Speed = 500 meters / 20 seconds Speed = 25 meters/second Therefore, the speed of the train is boxed{25} meters per second.

answer:To solve this problem, we need to analyze the relationship between the given conditions and understand the properties of inequalities. - We begin by testing if the condition "x geq 1 and y geq 2" is sufficient for "x + y geq 3". If both x and y satisfy their respective inequalities, by the property of the addition of inequalities, we have: x + y geq 1 + 2 = 3. This shows that if x geq 1 and y geq 2, then it must be the case that x + y geq 3. Therefore, the given condition is sufficient. - Now, we need to check if the condition is necessary. To do this, consider a counterexample. Let's take x = 0 and y = 4. It is clear that x + y = 0 + 4 = 4 geq 3. However, the condition x geq 1 is not satisfied. Thus, x + y geq 3 does not imply that both x geq 1 and y geq 2. From the definition of necessary and sufficient conditions, we conclude that "x geq 1 and y geq 2" is a sufficient but not necessary condition for "x + y geq 3". So the correct answer is: boxed{A}

answer:First, we find the prime factorization of each number: - 16 = 2^4 (since 16 is 2 times 2 times 2 times 2). - 24 = 2^3 cdot 3 (since 24 is 2times 2 times 2 times 3). - 45 = 3^2 cdot 5 (since 45 is 3 times 3 times 5). Next, identify the highest power of each prime factor present: - For 2, the highest power is 4 (from 16). - For 3, the highest power is 2 (from 45). - For 5, the highest power is 1 (from 45). The least common multiple (LCM) is obtained by multiplying these highest powers: LCM = 2^4 cdot 3^2 cdot 5 = 16 cdot 9 cdot 5 = 144 cdot 5 = 720. Conclusion: The least common multiple of 16, 24, and 45 is boxed{720}.