answer:To solve the given problem, we start by applying the product-to-sum identities to the given expression. Specifically, we use the identity 2cos a sin b = sin (a+b) - sin (a-b). This allows us to transform the original sum into a telescoping series. Let's break down the steps: 1. **Applying the Product-to-Sum Formula**: For each term 2cos(k^2a)sin(ka) in the sum, we apply the product-to-sum formula: [ 2cos(k^2a)sin(ka) = sin(k^2a + ka) - sin(k^2a - ka). ] This simplifies to sin(k(k+1)a) - sin((k-1)ka) by rearranging the terms. 2. **Telescoping the Series**: The sum now becomes [ sum_{k=1}^{n} [sin(k(k+1)a) - sin((k-1)ka)]. ] This series telescopes, meaning that most terms cancel out, leaving us with [ -sin(0) + sin(2a) - sin(2a) + sin(6a) - cdots - sin((n-1)na) + sin(n(n+1)a). ] Simplifying further, we find that all intermediate terms cancel, leaving [ -sin(0) + sin(n(n+1)a) = sin(n(n+1)a), ] since sin(0) = 0. 3. **Determining When the Sum is an Integer**: For the sum to be an integer, sin left(frac{n(n+1)pi}{2008}right) must be an integer. The sine function can only take integer values of -1, 0, or 1, which happens when its argument is an integer multiple of pi. Therefore, we need 2 cdot frac{n(n+1)}{2008} to be an integer. 4. **Finding the Smallest Positive Integer n**: The condition 2 cdot frac{n(n+1)}{2008} being an integer simplifies to 1004 = 2^2 cdot 251 dividing n(n+1). This implies that either n or n+1 must be a multiple of 251. The smallest such positive integer n that satisfies this condition is n = 251. Therefore, the smallest positive integer n for which the given expression is an integer is boxed{251}.

answer:To calculate the number of years the money was invested, we can use the formula for simple interest: Simple Interest (SI) = Principal (P) * Rate (R) * Time (T) / 100 We are given: Simple Interest (SI) = Rs. 155 Principal (P) = Rs. 810 Rate (R) = 4.783950617283951% We need to find the Time (T) in years. Rearranging the formula to solve for Time (T), we get: T = (SI * 100) / (P * R) Now, we can plug in the given values: T = (155 * 100) / (810 * 4.783950617283951) T = 15500 / (810 * 4.783950617283951) T = 15500 / 3874.9999999999995 T ≈ 4 Therefore, the money was invested for approximately boxed{4} years.

answer:If Jonas plans to put 3 shelves in the room and each shelf takes up 80 square feet of space, then the total space taken up by the shelves is: 3 shelves * 80 square feet/shelf = 240 square feet The room has a total of 400 square feet of space. If 240 square feet are taken up by the shelves, then the space reserved for the desk and walking space is: 400 square feet (total space) - 240 square feet (shelf space) = 160 square feet Jonas reserves boxed{160} square feet of space for the desk and walking space.

answer:First, we need to find the mean (overline{x}) of the data set: overline{x} = frac{1}{5}(9+10+9+7+10) = 9 Next, we calculate the variance (s^2) using the formula for variance: s^2 = frac{1}{5}left[(9-9)^2+(10-9)^2+(9-9)^2+(7-9)^2+(10-9)^2right] = frac{6}{5} So, the variance of the data set is boxed{frac{6}{5}}. To solve this problem, we simply apply the formulas for calculating the mean and variance of a data set. This is a basic question that tests one's understanding of these concepts.