Problem 175 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Sum the series \(x(x+y)+x^{2}\left(x^{2}+y^{2}\right)+x^{3}\left(x^{3}+y^{3}\right)+\cdots \cdots\) to \(n\) terms.

Step-by-Step Solution

Answer

The short version of the answer is: Sum: \(S_n = \sum_{k=1}^{n}x^{2k} + \sum_{k=1}^{n}x^k y^k\)

Step 1: Expand Each Term

To start the process, let's expand each term of the series: 1. First term: \(x(x+y) = x^2 + xy\) 2. Second term: \(x^2(x^2+y^2) = x^4 + x^2y^2\) 3. Third term: \(x^3(x^3+y^3) = x^6 + x^3y^3\) And so on.

Step 2: Recognize the Pattern

Observe that the powers of \(x\) in each term of the series are increasing by \(1\), first we have \(x^2\), then \(x^4\), and then \(x^6\). Also, in every term there is an extra factor containing terms with higher powers of \(x\) and some powers of \(y\). After expanding each term in the series, observe that the exponents of \(x\) and \(y\) are related; in the first term, the power of \(x\) and \(y\) are the same, in the second term, the power of \(x\) is twice the power of \(y\) and in the third term, the power of \(x\) is three times the power of \(y\).

Step 3: Write a General Term

Based on the pattern recognized, we can express the general term of the series as: A term: \(x^k(x^k+y^k)\), where \(k\) ranges from \(1\) to \(n\). Now, we can expand each term in the series: A term: \(x^{2k} + x^k y^k\), for \(k = 1, 2, 3, \dots, n\)

Step 4: Calculate the Sum

To find the sum of the series, it is now just a matter of adding up these terms for all values \(k\) from \(1\) to \(n\): Sum: \(S_n = \sum_{k=1}^{n}(x^{2k} + x^k y^k)\) Now, we can rewrite the sum as: Sum: \(S_n = \sum_{k=1}^{n}x^{2k} + \sum_{k=1}^{n}x^k y^k\)

Step 5: Interpret the Sum

This sum represents the sum of the series involving \(n\) terms, where the exponents of \(x\) and \(y\) change according to the pattern identified previously. By calculating this sum, we are able to obtain the result for any values of \(x\) and \(y\) and for any number of terms \(n\). The final expression for the sum is: Sum: \(S_n = \sum_{k=1}^{n}x^{2k} + \sum_{k=1}^{n}x^k y^k\)

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics used to denote repeated multiplication of a number by itself. For example, in the expression \(x^2\), "\(x\)" is the base and "\(2\)" is the exponent, indicating that \(x\) is multiplied by itself once: \(x \times x\). When there is no exponent shown, like \(x\), it is understood to be \(x^1\).In our series, exponents express how many times a variable is multiplied by itself. By using exponential form, we simplify the representation and manipulation of larger numbers and expressions.Working with Exponents in Mathematical Series

• They allow us to identify patterns easily, as in the case where successive terms show consistent powers of \(x\).
• Power rules such as \(x^a \times x^b = x^{a+b}\) help in combining terms efficiently.

Exponents not only make equations cleaner, but they are crucial in identifying the structure of series like the one in the exercise, which forms the basis for understanding the sum of series.

Sum of Series

A series is simply a sum of terms that follow a specific pattern or sequence. In mathematics, finding the sum of a series involves adding each term according to its sequence rules.Understanding the Sum of the Given SeriesThe series given in our exercise is a sum of terms of the form \(x^k(x^k + y^k)\) from \(k = 1\) to \(n\). Expanding each term yields terms like \(x^{2k}\) and \(x^k y^k\). Understanding how to sum these is essential:

• Separate the series into simpler components: \(\sum_{k=1}^{n} x^{2k}\) and \(\sum_{k=1}^{n} x^k y^k\).
• Each sub-series adds up individually following the powers defined for \(x\) and \(y\).

By structuring the sum in this way, the problem becomes more manageable, making it easier to handle complex calculations especially when \(n\) is large.

Pattern Recognition

Pattern recognition in mathematics allows us to make predictions and form equations based on observed regularities or trends. In the context of a series, recognizing a pattern helps in writing down general formulas that describe the whole sequence.Identifying Patterns in the SeriesIn the provided series, pattern recognition is key to determining the general formula:

• Notice the powers of \(x\): \(x^2, x^4, x^6, \ldots\) They form a sequence where each exponent is twice its index, \(2k\).
• Similarly, notice the relation of \(x\) to \(y\): The power of \(y\) in each term is \(k\), showing a direct relationship with the term's index.

Using these patterns, the series can be understood as a compilation of terms following a clear rule, allowing us to write a general formula: \(x^k(x^k + y^k)\) which simplifies terms and interpretations. Effective pattern recognition not only simplifies solving but also enhances understanding of underlying mathematical principles.