answer:1. **Understanding the Problem Setup:** We are given a product of numbers where each subsequent number in the product doubles the number of `9`s (e.g., (9), (99), (9999), and so on). Specifically, the number of 9's in the (i)-th term is (2^{i-1}). The product can be written as: [ a = 9 times 99 times 9999 times cdots times underbrace{99ldots99}_{2^{n-1}} ] We aim to find the sum of the digits of: [ a times underbrace{99ldots99}_{2^n} ] 2. **Applying Lemma 1:** - **Lemma 1:** Let (a) be a number with at most (m) decimal places, such that the last digit of (a) is not 0. Then, the sum of the digits of (underbrace{99ldots99}_{m} times a) is equal to (9m). - **Proof of Lemma 1:** If (a = a_{m-1}a_{m-2}ldots a_1a_0) is written in decimal representation with (a_0 neq 0), then: [ underbrace{99ldots99}_{m} times a = (10^m - 1)a ] Therefore, [ underbrace{99ldots99}_{m} times a = 10^{m}a - a ] 3. **Subtraction and Carries:** During the subtraction (10^m a - a), each digit contributes to a carry: - There will be m final carry operations for the last m digits. - Since (a_{0} neq 0), these carries will not propagate beyond the mth digit. 4. **Calculation of Sum of Digits:** - Since none of the digit carries cancel completely beyond the (m)th position, we tally each component: - (b_0 + a_0 = 10), - For (k) (1 to (m-1)), (b_k + a_k = 9 ), - The value (b_m + 1 = a_0), and for ( k geq m+1), ( b_k = a_{k-m} ). 5. **Summing up digits:** The total sum of the digits result in: [ sum_{k=0}^{2m-1} b_k = (10 - a_0) + sum_{k=1}^{m-1}(9 - a_k) + (a_0 - 0) + sum_{k=1}^{m-1} a_k ] Combining like terms, explicitly: [ sum_{k=0}^{2m-1} b_k = 9m ] 6. **Applying Lemma Beyond Proof:** Now for (a) and (underbrace{99ldots99}_{2^n}): - The number (a) is such that it does not end in zero and is less than (10^{2^n-1}) having at most (2^n-1) digits. - Applying Lemma 1 with (m = 2^n): 7. **Conclusion:** Sum of the digits: [ 9 times 2^n ] Hence, the final sum of the digits: [ boxed{9 cdot 2^n} ]

answer:To find out how many books Tim has, we can subtract the number of books Mike has from the total number of books they have together. So, Tim's books = Total books together - Mike's books Tim's books = 42 books - 20 books Tim's books = 22 books Therefore, Tim has boxed{22} books.

answer:Let's break down the information given step by step: 1. Initially, there are 16 cats in the house. 2. Two of these cats are white. 3. 25% of the cats are black. To find out how many black cats there are, we calculate 25% of 16: 25/100 * 16 = 4 black cats. 4. Half of the black cats left, so half of 4 is 2. This means 2 black cats left, leaving 2 black cats remaining in the house. 5. After the black cats left, 2 more white cats showed up. This means there are now 2 (original white cats) + 2 (new white cats) = 4 white cats in total. 6. The rest of the cats are grey. We initially had 16 cats, 4 of which are now white, and 2 are black. To find the number of grey cats initially, we subtract the white and black cats from the total: 16 (total cats) - 4 (white cats) - 2 (black cats) = 10 grey cats. 7. A grey cat that is three times older than the oldest white cat comes to the house. This is an additional grey cat, so we add 1 to the existing number of grey cats: 10 (initial grey cats) + 1 (new grey cat) = 11 grey cats. After all the occurrences, there are boxed{11} grey cats in the house.

answer:To determine the range of real numbers for k that ensures the function f(x)=4x^{2}-kx-8 is monotonically increasing on the interval left[5,+infty right), follow these steps: 1. Recognize that f(x)=4x^{2}-kx-8 is a quadratic function. The axis of symmetry of a quadratic function given by f(x)=ax^2+bx+c is x=-dfrac{b}{2a}. For our function, this translates to x=dfrac{k}{8} because a=4 and b=-k. 2. For f(x) to be monotonically increasing on left[5,+infty right), the axis of symmetry must be to the left of or at x=5. This condition ensures that the function is increasing for all x geq 5. Therefore, we have dfrac{k}{8} leq 5. 3. Solving the inequality dfrac{k}{8} leq 5 for k gives: [ dfrac{k}{8} leq 5 ] Multiplying both sides by 8 to isolate k, we obtain: [ k leq 40 ] 4. The inequality k leq 40 means that the range of k for which f(x) is monotonically increasing on the interval left[5,+infty right) is left(-infty, 40right]. Therefore, the correct answer, encapsulating the final answer with the specified notation, is boxed{B}.