Maths Sequences and Algebra PDF

I.1. A, C, F, J, O, - Question: What is the next letter in the sequence A, C, F, J, O? - Solution Process: This sequence follows a pattern based on the positions of the letters in the alphabet. The differences between consecutive letters increase by 1 each time: -C-A=2 -F-C=3 -J-F=4 -O-J=5 F...

I.1. A, C, F, J, O, - Question: What is the next letter in the sequence A, C, F, J, O? - Solution Process: This sequence follows a pattern based on the positions of the letters in the alphabet. The differences between consecutive letters increase by 1 each time: -C-A=2 -F-C=3 -J-F=4 -O-J=5 Following this pattern, the next difference should be 6. Therefore, the next letter is O + 6 = U. - Answer: U - Key Concepts/Tips: Identifying patterns in sequences is a crucial skill in mathematics. Look for differences, ratios, or other relationships between consecutive terms. I.2. 27, 20, 11, 0, -13, - Question: What is the next number in the sequence 27, 20, 11, 0, -13? - Solution Process: The sequence decreases by 7 each time: - 27 - 7 = 20 - 20 - 7 = 11 - 11 - 7 = 4 - 4 - 7 = -3 - -3 -7 = -10 - Answer: -10 - Key Concepts/Tips: Arithmetic sequences have a constant difference between consecutive terms. I.3. 0, 1, 16, 81, - Question: What is the next number in the sequence 0, 1, 16, 81? - Solution Process: This sequence represents the squares of consecutive powers of 2: - 0 = 0² = (2⁰)² - 1 = 1² = (2¹)² - 16 = 4² = (2²)² - 81 = 9² = (2³)² Following the pattern, the next term would be (2⁴)² = 16² = 256 - Answer: 256 - Key Concepts/Tips: Recognizing patterns involving powers and exponents is important. I.4. x + 3, 4x + 8, 7x + 13, - Question: What is the next term in the sequence x + 3, 4x + 8, 7x + 13? - Solution Process: The coefficients of x increase by 3 each time (1, 4, 7), and the constant terms increase by 5 each time (3, 8, 13). Therefore, the next term should be 10x + 18. - Answer: 10x + 18 - Key Concepts/Tips: Look for patterns in both the coefficients and the constant terms of algebraic expressions. I.5. 8, 6, 9/2, 27/8, - Question: What is the next number in the sequence 8, 6, 9/2, 27/8? - Solution Process: This sequence appears to be a geometric sequence where each term is multiplied by a decreasing fraction. Let's analyze the ratios: - 6/8 = 3/4 - (9/2)/6 = 3/4 - (27/8)/(9/2) = 3/4 Following this pattern, the next term is (27/8) * (3/4) = 81/32 - Answer: 81/32 - Key Concepts/Tips: Geometric sequences have a constant ratio between consecutive terms. II.1. Z⁻ ⊂Q - Question: Is the set of negative integers (Z⁻) a subset of the set of rational numbers (Q)? - Answer: TRUE - Explanation: All negative integers can be expressed as fractions (e.g., -3 = -3/1), making them rational numbers. II.2. N ⊂Z - Question: Is the set of natural numbers (N) a subset of the set of integers (Z)? - Answer: TRUE - Explanation: Natural numbers are a subset of integers; all natural numbers are integers. II.3. ∀x ∈ N, x² ≥ 0 - Question: For all x belonging to the set of natural numbers (N), is x² greater than or equal to 0? - Answer: TRUE - Explanation: The square of any natural number is always non-negative. I.4. ∃x ∈ R, ∀y ∈ I⁺ | y/x = 1 - Question: Does there exist a real number x such that for all positive integers y, y/x = 1? - Answer: FALSE - Explanation: If y/x = 1 for all positive integers y, then x would have to equal every positive integer simultaneously, which is impossible for a single real number x. II.5. ∀x ∈ Q, x² ≥ 1 - Question: For all x belonging to the set of rational numbers (Q), is x² greater than or equal to 1? - Answer: FALSE Explanation: Many rational numbers (e.g., 1/2) have squares less than 1. The statement is only true for rational numbers with an absolute value greater than or equal to 1. classifying it as Always True (AT), Always False (AF), or Sometimes True, Sometimes False (STSF), along with justifications: 1. The sum of even numbers is even. - Classification: AT (Always True) - Justification: The sum of any two even numbers is always even. This can be proven algebraically: Let 2m and 2n represent two even numbers (where m and n are integers). Their sum is 2m + 2n = 2(m + n). Since m + n is an integer, 2(m + n) is an even number. This extends to the sum of any number of even numbers. 2. The cube of an integer is odd. - Classification: STSF (Sometimes True, Sometimes False) - Justification: The cube of an odd integer is odd (e.g., 3³ = 27), but the cube of an even integer is even (e.g., 2³ = 8). 3. All perfect squares are even. - Classification: AF (Always False) - Justification: The squares of even numbers are even (e.g., 2² = 4, 4² = 16), but the squares of odd numbers are odd (e.g., 1² = 1, 3² = 9). 4. 2 - x < 3 - x - Classification: AF (Always False) - Justification: Subtracting 'x' from both sides leaves 2 < 3, which is always true. However, the original inequality implies that there are values of x that would make the inequality false. The inequality simplifies to 2 < 3 which is always true. Therefore, the statement is always false because it claims that 2 - x is less than 3 - x for all x. 5. f(2) = 3 - Classification: STSF (Sometimes True, Sometimes False) - Justification: This statement depends entirely on the definition of the function 'f'. There are many functions where f(2) could equal 3, and many where it would not. For example: - If f(x) = x + 1, then f(2) = 3 (True) - If f(x) = x², then f(2) = 4 (False) Let's analyze each statement in the image: 1. ℝ⊂ℤ This statement asserts that the set of real numbers (ℝ) is a subset of the set of integers (ℤ). This is false. Reason: Real numbers include all rational numbers (like fractions and integers) and irrational numbers (like π and √2). Integers are only whole numbers and their negatives. Since there are many real numbers that are not integers (irrational numbers, for instance), ℝcannot be a subset of ℤ. 2. ℕ⊂ℤ⁺ This statement says that the set of natural numbers (ℕ) is a subset of the set of positive integers (ℤ⁺). This is true. - Reason: Natural numbers are typically defined as the positive integers (1, 2, 3...). Therefore, every natural number is also a positive integer. 3. ∀x ∈ ℝ, x² > 0 This statement says that for all real numbers x, x² is greater than 0. This is false. - Reason: If x = 0, then x² = 0, which is not greater than 0. 4. ∀x, y ∈ ℝ, (x + y)² = x² + 2xy + y² This statement says that for all real numbers x and y, the square of their sum is equal to the sum of their squares plus twice their product. This is true. - Reason: This is the standard expansion of a binomial squared, a fundamental algebraic identity. 5. ∃m ∈ ℤ, m - n ≤ m + n This statement says there exists an integer m such that m - n is less than or equal to m + n. This is true. - Reason: This inequality simplifies to -n ≤ n, which is true for all non-negative integers n and for some negative integers n. 6. ∀x ∈ ℝ, √x³ = x This statement says that for all real numbers x, the cube root of x cubed is equal to x. This is false. - Reason: This is only true for non-negative real numbers. If x is negative, the cube root of a negative number is negative. For example, if x = -8, then √(-8)³ = -8, but √(-8)³ ≠ -8 (because cube root of a negative number is negative, but the square root is undefined). 7. ∃x, y ∈ ℚ, x - y = x This statement says there exist rational numbers x and y such that x - y = x. This is true. Reason: This simplifies to y = 0. Since 0 is a rational number, this statement is true. 8. ∀x ∈ ℤ, ∃n ∈ ℕ, x² = n² This statement says that for all integers x, there exists a natural number n such that x² = n². This is false. - Reason: This would imply that all integers have a non-negative square. However, negative integers do not have a square root that is a natural number. 9. ∀x ∈ ℝ, ∃y ∈ ℝ, x + y = 4 This statement says that for all real numbers x, there exists a real number y such that x + y = 4. This is true. - Reason: You can always find a y that satisfies this equation; simply solve for y: y = 4 - x. Since x is a real number, 4 - x will also be a real number. mathematical sentences, using at most one variable unless otherwise specified: 1.Five more than a number: x + 5 2.Two more than the eleventh power: x¹¹ + 2 3.Eight less than a number: x - 8 4.Four less than thrice a number: 3x - 4 5.Seven more than six times a number: 6x + 7 6. Three consecutive even integers: n, n + 2, n + 4 (Uses one variable, n, representing the first even integer) 7.Three consecutive odd integers: n, n + 2, n + 4 (Uses one variable, n, representing the first odd integer) 8.A fraction whose numerator is one more than twice its denominator: (2x + 1)/x (Uses one variable, x, representing the denominator) 9.Two numbers whose sum is 15: x + y = 15 (Uses two variables as explicitly stated) 10.The sum of the squares of x and y: x² + y² (Uses two variables as explicitly stated) 11.The cube root of the sum of x and y: ³√(x + y) (Uses two variables as explicitly stated) 12.The area of a square having a side of length S: S²