answer:Let's denote the number of dolls Peggy received between her birthday and Christmas as x. According to the information given, the total number of dolls Peggy has after receiving the dolls from her grandmother and the additional dolls between her birthday and Christmas is 51. We can set up the following equation: Peggy's original dolls + dolls from grandmother + birthday and Christmas dolls = total dolls 6 + 30 + x = 51 Now, let's solve for x: x = 51 - 6 - 30 x = 15 Peggy received 15 dolls between her birthday and Christmas. Now, we need to find the ratio of these 15 dolls to the 30 dolls her grandmother gave her. The ratio is: x (birthday and Christmas dolls) : grandmother's dolls 15 : 30 To simplify the ratio, we divide both numbers by the greatest common divisor, which is 15 in this case: 15/15 : 30/15 1 : 2 Therefore, the ratio of the dolls Peggy received between her birthday and Christmas to the dolls her grandmother gave her is boxed{1:2} .

answer:To compare 6^{0.7}, 0.7^6, and log_{0.7}6, we perform the analysis as follows: 1. First, let's compare 6^{0.7} and 0.7^6. We have 6^{0.7} which can be seen as (6^7)^{0.1}, and 0.7^6 as (0.7^6)^{1}. As 6^7 is much larger than 0.7^6, and since raising a number to a positive power preserves the order, we conclude that 6^{0.7} > 0.7^6. 2. Next, we compare 6^{0.7} and log_{0.7}6. We know that since 0.7 < 1, the function f(x) = log_{0.7}x is decreasing. This means that for any x > 1, log_{0.7}x will be negative. Given that 6 > 1, we have log_{0.7}6 < 0. On the other hand, 6^{0.7} is clearly positive (since any positive number raised to a positive power is positive). Hence, we have 6^{0.7} > log_{0.7}6. 3. Finally, we compare 0.7^6 and log_{0.7}6. We have already established that log_{0.7}6 < 0. Since 0.7^6 > 0 because any positive number raised to any power remains positive, we can determine that 0.7^6 > log_{0.7}6. Putting it all together, we find that log_{0.7}6 < 0 < 0.7^6 < 6^{0.7}. Therefore, the order is log_{0.7}6 < 0.7^6 < 6^{0.7}, which corresponds to option D. boxed{D}

answer:A Since the sum of the first 16 terms is greater than 0 (S_{16} > 0) and the sum of the first 17 terms is less than 0 (S_{17} < 0), it indicates that the 17th term is negative and significantly large in magnitude to make the sum of the first 17 terms negative. This implies that the sequence is decreasing. The maximum sum S_n occurs right before the sum starts to decrease, which is at n = 16. However, since the sum of the first 16 terms is positive and starts to decrease afterward, the maximum sum actually occurs at the last positive term, which is the 16th term. But considering the nature of an arithmetic sequence and the conditions given, the maximum sum occurs when n = 16 / 2 = 8. Therefore, the correct answer is boxed{text{A: 8}}.

answer:To solve the given problem, we start by applying the definition of the new operation to the inequality 3※x < 2. According to the operation's definition, we have: [ 3 text{※} x = 3x - 3 + x - 2 ] Substituting this into the inequality gives: [ 3x - 3 + x - 2 < 2 ] Combining like terms, we get: [ 4x - 5 < 2 ] Adding 5 to both sides of the inequality to isolate the term with x gives: [ 4x < 7 ] Dividing both sides by 4 to solve for x yields: [ x < frac{7}{4} ] Given that x is a positive integer, the only value of x that satisfies x < frac{7}{4} is x=1. Therefore, the positive integer solution of the inequality is: [ boxed{x=1} ]