Mathematics Answer Note #54

Question 1 17 , 52 , 87 , 122 find the two new numbers and determine if its geometric or arthimetic Answer To find the two new numbers in this sequence, we must first determine the pattern of the sequence. In this case, it appears to be increasing by 35 with each term (17 + 35 = 52, 52 + 35 = 87 and so on). Therefore, the next two terms in the sequence would be 122 + 35 = 157 and 157 + 35 = 192. Now, to determine if the sequence is geometric or arithmetic, we must look at the common difference or ratio between each term. In arithmetic sequences, the term difference is constant, while in geometric sequences the term ratio is constant. In this case, the term difference is not constant (35, 35, and 35), therefore it is not an arithmetic sequence. However, if we look at the ratio between each term (52/17 = 3, 87/52 = 3/2, and 122/87 = 3/2), we can see that it is constant at 3/2. Therefore, the sequence is geometric. Question 2 1,5,9,13 find the missing two numbers if the sequence is arthiematic adding or subtracting each time or geometric multiplying each time. then find f stands for function then find the new two numbers Answer The given sequence of 1, 5, 9, 13 follows an arithmetic pattern, where the difference between consecutive terms is the same. This pattern can be represented as x, x+4, x+8, x+12, with x being the first term. To find the missing two numbers, we can apply the arithmetic pattern to the last term (13) and find the next two terms in the sequence: 13+4= 17 17+4= 21 Therefore, the missing two numbers in the sequence are 17 and 21. To find the missing numbers using a function, we can represent the given sequence as f(x)= x+4, where f(x) represents the function and x represents the first term. Then, we can substitute 1, the first term, into the function to get f(1)= 1+4= 5, which is the second term in the sequence. Similarly, we can continue to substitute the resulting term back into the function to get the next term in the sequence. f(5)= 5+4= 9, f(9)= 9+4= 13, f(13)= 13+4= 17, f(17)= 17+4= 21 Therefore, the missing two numbers, 17 and 21, can also be found using the function f(x)= x+4. Question 3 A person sets out to cross a forest. On the first day, he completes 1/10th of the total distance. On the second day,he covers 2/3rd of the distance already travelled on the first day. He continues in this manner, alternating the days inwhich he travels 1/10th of the distance still to be covered on one day, with days on which he travels 2/3rd of the totaldistance already covered on the next day. At the end of the seventh day, he finds 45/2 km more will see the end ofhis journey. What is the total distance that he has to cover? Answer According to the given information, the person starts by completing 1/10th of the total distance on the first day. This means that on the second day, he has already traveled 1/10th of the total distance and then covers 2/3rd of it, which is 2/3 * 1/10 = 1/15th of the total distance. This pattern continues, with him completing 1/10th of the remaining distance on the odd-numbered days and 2/3rd of the already covered distance on the even-numbered days. Using this pattern, we can create a table to track the distance covered each day: | Day | Distance Covered | |-----|-----------------| | 1 | 1/10th | | 2 | 1/15th | | 3 | 1/20th | | 4 | 1/30th | | 5 | 1/45th | | 6 | 1/60th | | 7 | 1/90th | At the end of the seventh day, we know that the person still has to cover 45/2 km to reach the end of their journey. From the table, we can see that on the seventh day, they traveled 1/90th of the total distance, which is equal to 45/2 km. Therefore, the total distance to be covered is: (45/2) / (1/90) = 45 * (90/2) = 45 * 45 = 2025 kilometers. Question 4 The sum of three consecutive natural numbers is equal to 60. Find the product of the lowest and highest number. Answer To approach this problem, we need to understand what are natural numbers and how to work with consecutive numbers. Natural numbers are a part of the number system that starts from 1 and continues infinitely. These numbers are also known as counting numbers. Consecutive numbers are numbers that follow each other in order without any gaps or repetitions. For example, 3,4,5 are consecutive numbers. To find the product of the lowest and highest number in this problem, we can first express the three consecutive numbers as x, x+1, and x+2, starting from the lowest number. By adding these three numbers and equating the sum to 60, we get the equation: x + (x+1) + (x+2) = 60 Solving this equation, we get x = 19. Therefore, the three consecutive natural numbers are 19, 20, and 21. Now, to find the product of the lowest and highest number, we simply multiply 19 and 21. Therefore, the product is 399. Question 5 Sketch the graphs of the following functions on separate diagrams and state the range in each case. 1. f(x) = x-6, 2<x<4 2. f(x) = 2x-3, -1<=x<=3 3. f(x) = -4x+5, x<2 Answer To sketch the graphs of these functions and determine their range, we first need to understand the properties and behavior of each function. Function 1: f(x) = x-6, 2<x<4 This function is a linear equation with a slope of 1 and a y-intercept of -6. Since the given domain is 2<x<4, we only need to focus on the part of the graph between x=2 and x=4. To plot this, we can pick any two points within this interval, for example, x=3 and x=3.5 and then use the slope-intercept form to find the corresponding y values. When x=3, y = (3-6) = -3 When x=3.5, y = (3.5-6) = -2.5 Plotting these points on a graph paper and connecting them with a straight line gives us the graph of f(x) = x-6 for 2<x<4. The range of this function is all real numbers between -4 and -2, or in interval notation: (-4, -2). Function 2: f(x) = 2x-3, -1<=x<=3 Similar to the first function, this is also a linear equation with a slope of 2 and a yintercept of -3. The given domain here is -1<=x<=3, so we plot the points between x=-1 and x=3. Using the same method as above, we can find the corresponding y values by substituting the given x values in the equation. When x=-1, y = (2*(-1)-3) = -5 When x=3, y = (2*3-3) = 3 Plotting these points and connecting them with a line, we get the graph of f(x) = 2x-3 for -1<=x<=3. The range of this function is again all real numbers, but this time between -5 and 3, or in interval notation: (-5, 3). Function 3: f(x) = -4x+5, x<2 This function is also a linear equation with a slope of -4 and a y-intercept of 5. However, the given domain here is x<2, indicating that we only need to focus on the part of the graph to the left of x=2. Similar to the previous two functions, we can find the corresponding y values by picking any two x values, for example, x=1 and x=0.5, and then plotting them on a graph paper. When x=1, y = (-4*1+5) = 1 When x=0.5, y = (-4*0.5+5) = 2 Connecting these points with a line gives us the graph of f(x) = -4x+5 for x<2. The range of this function is all real numbers, but this time to the left of x=2, so in interval notation it would be: (-∞, 5).