2. 1 Given The Quadratic Number Pattern 1; X; 1; -2; Y; ... ; -322. 2. 1. 1 Solve For X And Y. 3. 1. 2 Calculate The Number Of Terms In This Pattern. 4. 2 Given S_n = 5n - 3, Determine T_{34}. 5. 3 In An Arithmetic Sequence, The Tenth Term Is 28. The Sum Of Terms 5 To 20 Is 100. Find The First Term And The Common Difference.

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In the realm of mathematics, number patterns hold a captivating allure. They present us with sequences of numbers that follow specific rules, challenging us to decipher the underlying logic and predict future terms. Among these patterns, quadratic number patterns and arithmetic sequences stand out as fundamental concepts with wide-ranging applications. In this comprehensive exploration, we will delve into the intricacies of these patterns, unraveling their properties and mastering the techniques to solve related problems. We will dissect a specific quadratic number pattern, determine unknown terms, calculate the number of terms, and explore the characteristics of arithmetic sequences, ultimately equipping you with the skills to conquer these mathematical puzzles.

2.1 Cracking the Code of a Quadratic Number Pattern

Let's embark on our journey by examining the quadratic number pattern: 1; x; 1; -2; y; ... ; -322. A quadratic number pattern is a sequence where the second difference between consecutive terms is constant. This unique characteristic distinguishes it from linear patterns (where the first difference is constant) and other more complex patterns. Our mission is to solve for the unknown terms, 'x' and 'y,' and determine the total number of terms in this intriguing sequence.

2.1.1 Unmasking the Unknowns: Solving for x and y

To solve for 'x' and 'y,' we must first understand the underlying structure of a quadratic pattern. A quadratic pattern can be represented by the general formula: T_n = an^2 + bn + c, where T_n is the nth term, and 'a,' 'b,' and 'c' are constants. Our goal is to find the values of these constants using the given terms in the sequence. Let's meticulously analyze the provided terms to extract the necessary information.

  1. Formulating Equations: We can substitute the known terms and their positions into the general formula to create a system of equations. Let's start with the first term (T_1 = 1): 1 = a(1)^2 + b(1) + c, which simplifies to a + b + c = 1. Next, we consider the third term (T_3 = 1): 1 = a(3)^2 + b(3) + c, which simplifies to 9a + 3b + c = 1. Finally, we use the fourth term (T_4 = -2): -2 = a(4)^2 + b(4) + c, which simplifies to 16a + 4b + c = -2. Now we have a system of three equations with three unknowns:

    • a + b + c = 1
    • 9a + 3b + c = 1
    • 16a + 4b + c = -2

  2. Solving the System: There are several methods to solve this system of equations, such as substitution, elimination, or matrix methods. Let's employ the elimination method. Subtract the first equation from the second equation to eliminate 'c': (9a + 3b + c) - (a + b + c) = 1 - 1, which simplifies to 8a + 2b = 0. Next, subtract the second equation from the third equation to eliminate 'c': (16a + 4b + c) - (9a + 3b + c) = -2 - 1, which simplifies to 7a + b = -3. Now we have a simpler system of two equations with two unknowns:

    • 8a + 2b = 0
    • 7a + b = -3

    We can further simplify the first equation by dividing by 2: 4a + b = 0. Now, subtract this equation from the second equation: (7a + b) - (4a + b) = -3 - 0, which simplifies to 3a = -3. Therefore, a = -1. Substitute the value of 'a' back into the equation 4a + b = 0: 4(-1) + b = 0, which gives us b = 4. Finally, substitute the values of 'a' and 'b' into the equation a + b + c = 1: -1 + 4 + c = 1, which gives us c = -2. We have successfully determined the constants: a = -1, b = 4, and c = -2.

  3. Finding x and y: Now that we have the general formula for the quadratic pattern, T_n = -n^2 + 4n - 2, we can find 'x' and 'y.' The term 'x' is the second term (T_2): x = -(2)^2 + 4(2) - 2 = -4 + 8 - 2 = 2. The term 'y' is the fifth term (T_5): y = -(5)^2 + 4(5) - 2 = -25 + 20 - 2 = -7. Therefore, x = 2 and y = -7. We have successfully unveiled the hidden values within the quadratic pattern!

2.1.2 Counting the Terms: Calculating the Number of Terms

Our next challenge is to determine the number of terms in this pattern. We know the last term is -322. To find the position of this term (n), we can set the general formula equal to -322 and solve for 'n': -n^2 + 4n - 2 = -322. This equation can be rearranged into a quadratic equation: n^2 - 4n - 320 = 0. Let's employ our algebraic prowess to solve this equation.

  1. Solving the Quadratic Equation: We can solve this quadratic equation using factoring, the quadratic formula, or completing the square. Let's use factoring. We need to find two numbers that multiply to -320 and add up to -4. These numbers are -20 and 16. Therefore, we can factor the equation as (n - 20)(n + 16) = 0. This gives us two possible solutions: n = 20 or n = -16. Since the number of terms cannot be negative, we discard n = -16. Therefore, n = 20. We have successfully determined that there are 20 terms in this quadratic pattern.

2.2 Decoding Arithmetic Sequences: Finding T_{34} from S_n

Let's shift our focus to arithmetic sequences. An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference. We are given the sum of the first 'n' terms of an arithmetic sequence, S_n = 5n - 3, and our mission is to determine the 34th term, T_{34}. To accomplish this, we need to leverage the relationship between the sum of an arithmetic series and its individual terms.

  1. Understanding the Relationship: The sum of the first 'n' terms of an arithmetic sequence is given by the formula: S_n = (n/2)[2a + (n-1)d], where 'a' is the first term and 'd' is the common difference. However, we are given S_n in a different form, so we need to find an alternative approach. We can use the fact that T_n = S_n - S_{n-1} to find the nth term. Let's apply this to our problem.

  2. **Finding T_34}** To find T_{34, we need to calculate S_34} and S_{33}. S_{34} = 5(34) - 3 = 170 - 3 = 167. S_{33} = 5(33) - 3 = 165 - 3 = 162. Now we can find T_{34} T_{34 = S_{34} - S_{33} = 167 - 162 = 5. Therefore, the 34th term of the arithmetic sequence is 5. We have successfully deciphered the term using the given sum formula!

2.3 Unveiling Arithmetic Sequence Properties: Determining Terms from Given Information

Our final challenge involves an arithmetic sequence where we are given the tenth term (T_{10} = 28) and a relationship between the sum of terms. We are told that the sum of term 5 to term 20 is 100. Our goal is to extract further information about this sequence, such as the first term and the common difference. Let's embark on this mathematical endeavor.

  1. Leveraging Arithmetic Sequence Formulas: Recall the general formula for the nth term of an arithmetic sequence: T_n = a + (n-1)d, where 'a' is the first term and 'd' is the common difference. We also need the formula for the sum of an arithmetic series: S_n = (n/2)[2a + (n-1)d]. We can adapt this formula to find the sum of terms from term 'm' to term 'n': S_{m to n} = S_n - S_{m-1}. Let's apply these formulas to the given information.

  2. Formulating Equations: We know T_10} = 28, so we can write the equation 28 = a + (10-1)d, which simplifies to a + 9d = 28. We also know that the sum of terms 5 to 20 is 100. This can be expressed as S_{20 - S_{4} = 100. Let's expand this using the sum formula: (20/2)[2a + (20-1)d] - (4/2)[2a + (4-1)d] = 100, which simplifies to 10[2a + 19d] - 2[2a + 3d] = 100. Further simplification yields: 20a + 190d - 4a - 6d = 100, which simplifies to 16a + 184d = 100. We can divide this equation by 4 to get: 4a + 46d = 25. Now we have a system of two equations with two unknowns:

    • a + 9d = 28
    • 4a + 46d = 25

  3. Solving the System: Let's solve this system using the elimination method. Multiply the first equation by -4: -4a - 36d = -112. Add this equation to the second equation: (4a + 46d) + (-4a - 36d) = 25 + (-112), which simplifies to 10d = -87. Therefore, d = -8.7. Substitute the value of 'd' back into the equation a + 9d = 28: a + 9(-8.7) = 28, which gives us a - 78.3 = 28. Therefore, a = 106.3. We have successfully determined the first term (a = 106.3) and the common difference (d = -8.7) of this arithmetic sequence.

In this comprehensive exploration, we have delved into the fascinating world of quadratic number patterns and arithmetic sequences. We have mastered the techniques to solve for unknown terms in quadratic patterns, calculate the number of terms, and decipher the properties of arithmetic sequences using given information. By understanding the underlying formulas and applying systematic problem-solving approaches, we have successfully conquered the challenges presented by these mathematical puzzles. These skills will undoubtedly serve as a strong foundation for further mathematical endeavors and real-world applications.