Mathematics Answer Note #58

Question 1 The term-to-term rule for a sequence is 'subtract 1.5'. The 1st term of the sequence is 40. Find the first negative term in the sequence. Answer According to the given term-to-term rule of subtracting 1.5, the sequence would look like this: 40, 40-1.5, 40-3, 40-4.5, 40-6, 40-7.5,... To find the first negative term in the sequence, we need to find the point where the value becomes negative. From the sequence above, we can see that the value becomes negative at the fourth term, which is 40-6 = 34. Therefore, the first negative term in the sequence is 34. Step-by-step breakdown: 1. Understand the term-to-term rule: In a sequence, the term-to-term rule defines the relationship between each term in the sequence. In this case, the rule is 'subtract 1.5', meaning each term in the sequence will be 1.5 less than the previous term. 2. Write out the sequence: Starting with the first term, which is given as 40, we can use the term-to-term rule to write out the rest of the sequence. The sequence will be: 40, 401.5, 40-3, 40-4.5, 40-6, 40-7.5,... 3. Find the first negative term: We can see from the sequence that the first negative term occurs at the fourth term, which is 40-6 = 34. This is the first negative term in the sequence. Question 2 If a sum of money amounts to Rs 1326 at 5% simple interest in 6 years when will it amount to Rs 1530 at the same rate Answer To solve this problem, we will first use the simple interest formula, I = P*r*t, where I is the interest earned, P is the principal amount, r is the interest rate, and t is the time in years. Plugging in the given values, we get: I = (1326)(0.05)(6) I = 396.9 Since we want to find out when the sum of money reaches Rs 1530, we need to add the interest earned (396.9) to the original principal (1326) to get the new amount, which is Rs 1722.9. Next, we can use the same formula again to find out the time it takes to reach Rs 1722.9 at 5% simple interest. This time, we know the interest earned (396.9), the principal (1326), and the new amount (1722.9), so we can solve for t (time). I = (1722.9 - 1326)(0.05)(t) 396.9 = (396.9)(t) t = 1 year Therefore, the sum of money will reach Rs 1530 at 5% simple interest in 7 years. Question 3 Find the general solution for the series: 2, 1, 3, 4, 7, 11, 18, … (2 and 1 are the initial values of the series.). Also, find the value of u 20 [Ans: u k = , u 20 = 15127 Answer To find the general solution for the given series, we first need to determine the pattern and sequence that it follows. Upon careful observation, we can see that the series is generated by adding the previous two terms in the series, starting with the initial values of 2 and 1. This type of series is known as a Fibonacci sequence, where each number is the sum of the two previous numbers. To find the formula for the nth term in the series, we can use the following formula: u_n = u_n-1 + u_n-2 where u_n represents the nth term in the series. In this case, the initial values of u₀ = 2 and u₁ = 1, so the formula becomes: u_n = u_n-1 + u_n-2 = u_n-2 + u_n-3 + u_n-2 = 2u_n-2 + u_n-3 This gives us the following sequence: 2, 1, 3, 4, 7, 11, 18, 29, … Using this formula, we can find the value of the 20th term, u₂₀, by substituting n=20 in the above formula: u₂₀ = 2u_20-2 + u_20-3 = 2u₁₈ + u₁₇ To find the value of u₁₈, we can continue to use the formula until we reach the initial values, u₀ and u₁. Or, we can also use the fact that the nth term in a Fibonacci sequence can be expressed as: u_n = [φⁿ - (1-φ)ⁿ] / √5 where φ = (1+√5)/2 is the golden ratio. Using this formula, we can directly find the value of u₁₈ as: u₁₈ = [φ¹⁸ - (1-φ)¹⁸] / √5 = [1346269 - (-1346269)] / √5 = 1346269 / √5 = 104911/5√5 Using the same formula, we can find the value of u₁₇ as: u₁₇ = [φ¹⁷ - (1-φ)¹⁷] / √5 = [832040 - (-832040)] / √5 = 832040 / √5 Substituting these values in our above equation, we get: u₂₀ = 2u₁₈ + u₁₇ = 2(104911/5√5) + 832040 / √5 = (2*104911 + 832040)/√5 = (209822 + 832040)/√5 = 1049862/√5 Therefore, the value of u₂₀, the 20th term in the series, is approximately equal to 15127. To confirm this, we can use the formula we derived earlier, i.e. u_n = [φⁿ - (1-φ)ⁿ] / √5, to calculate the exact value: u₂₀ = [φ²⁰ - (1-φ)²⁰] / √5 = [6765 - (-6765)] / √5 = 6765 / √5 = 10526.937 Hence, the value of u_{20</ Question 4 What can Leo conclude about the volumes of the two toys? Answer Based on the given information, Leo can conclude that the volume of the toy car is smaller than the volume of the toy boat. This can be inferred from the difference in the water level rise when each object is placed separately in the container of water. The toy car caused the water level to rise by 20 millimeters, while the toy boat caused a higher rise of 30 millimeters. This indicates that the toy boat has a larger volume, as it displaces more water than the toy car. To further support this conclusion, Archimedes' Principle can be applied. According to this principle, the buoyant force acting on an object is equal to the weight of the water displaced by the object. In this case, since both toys are made of plastic, they have the same density and weight. Therefore, the difference in the water level rise can be attributed to the difference in volume. Question 5 Can a prime number be a multiple of any other number except itself? Explain your answer why or why not Answer According to the fundamental properties of prime numbers, they can only be evenly divisible by 1 and itself, making them inherently different from other numbers. This raises the question, can a prime number be a multiple of any other number except itself? The answer is no, and in this response, we will delve into the reasons why. First and foremost, let us establish the definition of a prime number. A prime number is a positive integer that has exactly two positive divisors, 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, etc. One key property of prime numbers is that they cannot be expressed as a product of smaller numbers, making them unique and essential in the field of mathematics. Thus, by definition, a prime number cannot be a multiple of any other number except itself. Apart from the definition of prime numbers, there are various mathematical proofs that support this statement. One such proof is the Fundamental Theorem of Arithmetic, which states that every positive integer greater than 1 can be expressed as a unique product of prime numbers. This theorem serves as the basis for the prime factorization method, which is used to determine the prime factors of any given number. For instance, if we take a prime number like 19, we can see that it can only be expressed as 1 x 19 or 19 x 1, making it a multiple of itself only. Furthermore, we can also employ a proof by contradiction to show that a prime number cannot be a multiple of any other number except itself. Let us assume that there exists a prime number "p" that is a multiple of some number "m" other than 1 and itself. This would mean that "p" is divisible by "m," and hence, can be written as p = m x k, where k is some integer. However, this contradicts the definition of a prime number, where it is stated that a prime number can only have two factors, 1 and itself. Therefore, our assumption is false, and it can be concluded that a prime number cannot be a multiple of any other number except itself. To further support our argument, let us also look at the concept of coprimes or relatively prime numbers. Two numbers are said to be coprime or relatively prime if they do not have any common factors other than 1. Prime numbers, by definition, are coprime to every other number, except for itself. This again illustrates the uniqueness of prime numbers and how they cannot be multiples of any other number. In conclusion, prime numbers, by their very definition and various mathematical proofs, cannot be multiples of any other number except themselves. They play a crucial role in mathematics and have been a subject of great interest for mathematicians for centuries. As a student, it is essential to have a clear understanding of prime numbers and their properties to excel in the field of mathematics.}