Determining the formula for the nth term of a sequence is a fundamental problem in mathematics, often encountered in calculus, discrete mathematics, and number theory. This task involves identifying a pattern within the sequence and expressing it as a mathematical function of n, where n represents the position of the term in the sequence (e.g., n=1 for the first term, n=2 for the second term, and so on). The given sequence, 3, 10, 29, 66, 127, ..., presents an interesting challenge. At first glance, the differences between consecutive terms don't appear to follow a simple arithmetic or geometric progression. However, a deeper analysis reveals a hidden structure related to powers of integers. This article aims to meticulously dissect this sequence, identify the underlying pattern, and construct a precise formula for its nth term. We will explore various techniques for pattern recognition and formula derivation, providing a clear, step-by-step approach that can be applied to similar problems. This process will not only uncover the specific formula for this sequence but also enhance your problem-solving skills in mathematics, enabling you to tackle other complex sequences with confidence. Through this exploration, we will delve into the fascinating world of sequence analysis, discovering how mathematical patterns can be elegantly expressed using formulas. The ability to find these formulas is crucial in various applications, from predicting future terms in a sequence to modeling real-world phenomena that exhibit sequential behavior. This journey into the heart of sequence analysis will provide a solid foundation for further studies in mathematics and related fields, demonstrating the power and beauty of mathematical reasoning.
Analyzing the Sequence: 3, 10, 29, 66, 127, ...
When analyzing a sequence, the first step is to examine the differences between consecutive terms. This approach often reveals a pattern that might not be immediately apparent. Let's apply this method to our sequence: 3, 10, 29, 66, 127, ... The differences between consecutive terms are: 10 - 3 = 7, 29 - 10 = 19, 66 - 29 = 37, and 127 - 66 = 61. The resulting sequence of differences is 7, 19, 37, 61, .... While this sequence is not constant, it suggests that the original sequence is not a simple arithmetic progression. To further investigate, let's calculate the differences between the terms in this new sequence: 19 - 7 = 12, 37 - 19 = 18, and 61 - 37 = 24. Now we have a sequence of differences: 12, 18, 24, .... This sequence is an arithmetic progression with a common difference of 6, indicating that the original sequence might be related to a cubic function (a polynomial of degree 3). The reason a constant second difference suggests a cubic relationship is rooted in the properties of polynomials. When you take the first difference of a polynomial sequence, you reduce the degree of the polynomial by one. Taking the second difference reduces the degree by another one. If the second difference is constant, it implies that the original sequence was generated by a polynomial of degree 3. This method of analyzing differences is a powerful tool in sequence analysis, allowing us to identify the underlying mathematical structure. In our case, the constant second difference points towards a cubic function as the generator of the sequence. From this point, we can hypothesize a general form for the nth term and use the given terms to determine the specific coefficients. This process often involves setting up a system of equations and solving for the unknowns. The successful application of this technique demonstrates the importance of pattern recognition and the ability to translate observed patterns into mathematical expressions. This skill is not only valuable in solving sequence problems but also in modeling real-world phenomena that exhibit similar patterns.
Identifying the Pattern and Hypothesizing a Formula
Given that the second differences are constant, we hypothesize that the formula for the nth term is a cubic polynomial of the form: an = An^3 + Bn^2 + Cn + D where A, B, C, and D are constants that we need to determine. To find these constants, we will use the first four terms of the sequence: 3, 10, 29, and 66. By substituting n = 1, 2, 3, and 4 into the equation, we can create a system of four linear equations with four unknowns. This system will allow us to solve for A, B, C, and D, thus defining the specific cubic polynomial that generates our sequence. Let's start by plugging in n = 1: a1 = A(1)^3 + B(1)^2 + C(1) + D = A + B + C + D = 3 Next, for n = 2: a2 = A(2)^3 + B(2)^2 + C(2) + D = 8A + 4B + 2C + D = 10 For n = 3: a3 = A(3)^3 + B(3)^2 + C(3) + D = 27A + 9B + 3C + D = 29 And finally, for n = 4: a4 = A(4)^3 + B(4)^2 + C(4) + D = 64A + 16B + 4C + D = 66 Now we have a system of four linear equations: 1. A + B + C + D = 3 2. 8A + 4B + 2C + D = 10 3. 27A + 9B + 3C + D = 29 4. 64A + 16B + 4C + D = 66 Solving this system of equations is a crucial step in finding the formula. There are several methods to solve such systems, including substitution, elimination, and matrix methods. Each method involves manipulating the equations to isolate the variables and find their values. The choice of method often depends on the specific equations and personal preference. Once we solve for A, B, C, and D, we will have the complete formula for the nth term of the sequence. This formula will allow us to calculate any term in the sequence, no matter how far out, and provides a concise mathematical representation of the sequence's pattern. The ability to derive such formulas is a testament to the power of mathematical modeling and the beauty of expressing complex patterns in a simple, elegant form.
Solving the System of Equations
To solve the system of equations, we can use the method of elimination. This involves subtracting equations from each other to eliminate variables one at a time. Let's rewrite our system of equations for clarity: 1. A + B + C + D = 3 2. 8A + 4B + 2C + D = 10 3. 27A + 9B + 3C + D = 29 4. 64A + 16B + 4C + D = 66 First, subtract equation (1) from equation (2): (8A + 4B + 2C + D) - (A + B + C + D) = 10 - 3 7A + 3B + C = 7 (5) Next, subtract equation (2) from equation (3): (27A + 9B + 3C + D) - (8A + 4B + 2C + D) = 29 - 10 19A + 5B + C = 19 (6) Then, subtract equation (3) from equation (4): (64A + 16B + 4C + D) - (27A + 9B + 3C + D) = 66 - 29 37A + 7B + C = 37 (7) Now we have a new system of three equations with three unknowns: 5. 7A + 3B + C = 7 6. 19A + 5B + C = 19 7. 37A + 7B + C = 37 Subtract equation (5) from equation (6): (19A + 5B + C) - (7A + 3B + C) = 19 - 7 12A + 2B = 12 6A + B = 6 (8) Subtract equation (6) from equation (7): (37A + 7B + C) - (19A + 5B + C) = 37 - 19 18A + 2B = 18 9A + B = 9 (9) Now we have two equations with two unknowns: 8. 6A + B = 6 9. 9A + B = 9 Subtract equation (8) from equation (9): (9A + B) - (6A + B) = 9 - 6 3A = 3 A = 1 Substitute A = 1 into equation (8): 6(1) + B = 6 6 + B = 6 B = 0 Substitute A = 1 and B = 0 into equation (5): 7(1) + 3(0) + C = 7 7 + C = 7 C = 0 Substitute A = 1, B = 0, and C = 0 into equation (1): 1 + 0 + 0 + D = 3 D = 2 Therefore, A = 1, B = 0, C = 0, and D = 2. This completes the process of solving the system of equations. Each step, from the initial elimination of variables to the final substitution, is crucial in arriving at the correct values. The systematic approach used here demonstrates the power of algebraic manipulation in solving complex problems. With the values of A, B, C, and D determined, we are now ready to construct the formula for the nth term of the sequence, providing a concise and accurate representation of the pattern.
Constructing the Formula and Verification
Now that we have found the values of A, B, C, and D, we can construct the formula for the nth term of the sequence. Recall that we hypothesized a cubic polynomial of the form: an = An^3 + Bn^2 + Cn + D Substituting the values A = 1, B = 0, C = 0, and D = 2, we get: an = 1*n^3 + 0*n^2 + 0*n + 2 Simplifying, the formula becomes: an = n^3 + 2 This is the formula that we believe generates the sequence 3, 10, 29, 66, 127, ... To verify that this formula is correct, we can test it with the given terms of the sequence. Let's calculate the first few terms using our formula: For n = 1: a1 = (1)^3 + 2 = 1 + 2 = 3 For n = 2: a2 = (2)^3 + 2 = 8 + 2 = 10 For n = 3: a3 = (3)^3 + 2 = 27 + 2 = 29 For n = 4: a4 = (4)^3 + 2 = 64 + 2 = 66 For n = 5: a5 = (5)^3 + 2 = 125 + 2 = 127 The formula generates the correct terms for n = 1, 2, 3, 4, and 5, which strongly suggests that it is indeed the correct formula for the sequence. Verification is a crucial step in the problem-solving process, as it ensures that our solution is accurate and reliable. By testing the formula with known values, we gain confidence in its correctness. In this case, the formula an = n^3 + 2 accurately represents the sequence, providing a concise and elegant mathematical expression for the underlying pattern. This formula can be used to calculate any term in the sequence, demonstrating the power of mathematical modeling in capturing and representing complex patterns. The successful derivation and verification of this formula showcase the importance of systematic problem-solving techniques, from initial pattern recognition to the final validation of the solution.
Final Answer
The formula for the nth term of the sequence 3, 10, 29, 66, 127, ... is: an = n^3 + 2 for n ≥ 1. This formula concisely captures the pattern exhibited by the sequence, allowing us to calculate any term in the sequence by simply substituting the desired value of n. The process of deriving this formula involved several key steps, including analyzing the differences between consecutive terms, hypothesizing a general form for the formula (in this case, a cubic polynomial), setting up and solving a system of equations, and finally, verifying the formula with the given terms of the sequence. Each of these steps is essential in the problem-solving process, and the successful application of these techniques demonstrates the power of mathematical reasoning. The ability to find such formulas is not only valuable in mathematics but also in various other fields where patterns and sequences arise, such as computer science, physics, and engineering. The formula an = n^3 + 2 provides a clear and elegant representation of the sequence, highlighting the beauty and power of mathematical expressions. This final answer is the culmination of a systematic and rigorous analysis, showcasing the importance of precision and attention to detail in mathematical problem-solving. The journey from the initial sequence to the final formula is a testament to the power of mathematical thinking and the ability to extract patterns from seemingly complex data.
