Question

Rewrite the following in \(\sum\) notation: (a) \(x_{1}\left(x_{1}-1\right)+2 x_{2}\left(x_{2}-1\right)+3 x_{3}\left(x_{3}-1\right)\) (b) \(a_{2}\left(x_{3}+2\right)+a_{3}\left(x_{4}+3\right)+a_{4}\left(x_{5}+4\right)\) (c) \(\frac{1}{x}+\frac{1}{x^{2}}+\dots+\frac{1}{x^{n}} \quad(x \neq 0)\) (d) \(1+\frac{1}{x}+\frac{1}{x^{2}}+\cdots-\frac{1}{x^{n}} \quad(x \neq 0)\)

Short answer

(a) \( \sum_{n=1}^{3} n x_n (x_n - 1) \); (b) \( \sum_{n=2}^{4} a_n (x_{n+1} + n) \); (c) \( \sum_{k=1}^{n} \frac{1}{x^k} \); (d) \( 1 + \sum_{k=1}^{n} \frac{1}{x^k} \).\)

Step-by-step solution

Identify the pattern in expression (a)

The expression \(x_{1}(x_{1}-1) + 2x_{2}(x_{2}-1) + 3x_{3}(x_{3}-1)\) shows a pattern where each term can be expressed as \(nx_n(x_n-1)\), where \(n\) takes values from 1 to 3.

Write expression (a) in summation notation

Using the identified pattern, the expression can be rewritten using summation notation as: \[ \sum_{n=1}^{3} n imes x_n (x_n - 1) \] where \(n\) is the index variable iterating from 1 to 3.

Identify the pattern in expression (b)

The expression \(a_{2}(x_{3} + 2) + a_{3}(x_{4} + 3) + a_{4}(x_{5} + 4)\) follows a pattern \(a_{n}(x_{n+1} + n)\), where \(n\) starts at 2 and ends at 4.

Write expression (b) in summation notation

The pattern allows us to express it as: \[ \sum_{n=2}^{4} a_n (x_{n+1} + n) \] where \(n\) varies from 2 to 4.

Identify the pattern in expression (c)

The expression \(\frac{1}{x} + \frac{1}{x^{2}} + \cdots + \frac{1}{x^{n}}\) follows the pattern of a geometric series \(\frac{1}{x^k}\) where \(k\) starts at 1 and ends at \(n\).

Write expression (c) in summation notation

The expression in summation form is: \[ \sum_{k=1}^{n} \frac{1}{x^k} \] where \(k\) is the index variable iterating from 1 to \(n\).

Identify pattern in expression (d)

The expression \(1 + \frac{1}{x} + \frac{1}{x^{2}} + \cdots + \frac{1}{x^{n}}\) adds an additional term of 1 to the series seen in (c).

Write expression (d) in summation notation

The expression can be rewritten as:\[ 1 + \sum_{k=1}^{n} \frac{1}{x^k} \] indicating the series begins with 1 followed by a series starting at \(\frac{1}{x}\).

Key concepts

Index Variable

In mathematics, the index variable, often denoted by symbols like \(n\) or \(k\), plays a crucial role in summation notation. It's like the variable that tells you where to start and end when adding up a series of terms. Imagine you're counting steps on a staircase; the index variable marks which step you're on.
For example, in the expression \(\sum_{n=1}^{3} n \times x_n (x_n - 1)\), the index variable is \(n\). It starts at 1 and increases step-by-step until it reaches 3. Each number refers to a specific term in the series, such as \(1 \times x_1 (x_1 - 1)\), then \(2 \times x_2 (x_2 - 1)\), and so on.
The index variable is essential because:

• It helps define the limits of a series.
• It clearly states the starting and ending points.
• It aids in maintaining consistency across terms in a series.

By understanding how the index variable operates within summations, you can smoothly manage complex expressions and see the overarching patterns.

Geometric Series

A geometric series is a particular type of series in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This kind of series is pervasive in mathematics because it efficiently models exponential growth or decay.
Understanding how a geometric series functions is vital for solving problems like the one in expression (c) from the original exercise. Let's break down the given series: \(\frac{1}{x} + \frac{1}{x^2} + \dots + \frac{1}{x^n}\).
Here, the series is geometric because each term can be expressed as \(\frac{1}{x^k}\), where \(k\) is the index variable that changes from 1 to \(n\). The fixed common ratio for this series is \(\frac{1}{x}\), indicating how each term relates multiplicatively to its predecessor.
Key points about a geometric series:

• The first term and the common ratio are enough to describe the entire sequence.
• Geometric series can rapidly increase or decrease, depending on the value of the common ratio.
• Knowing how to write a geometric series in summation helps in simplifying complex processes.

Geometric series appear in various fields, from finance to physics, illustrating patterns that recur across disciplines.

Mathematical Patterns

Recognizing and leveraging mathematical patterns is a condensed way to simplify and handle complex expressions. They assist both in predicting the progression of terms and in transforming these terms into a more manageable mathematical form like summation notation.
For instance, in expression (d) from the exercise, the sequence \(1 + \frac{1}{x} + \frac{1}{x^2} + \cdots + \frac{1}{x^n}\) demonstrates an obvious pattern beginning with 1 followed by terms that form a geometric series. It helps make problems easier to solve when you can spot such familiar patterns.
The benefits of identifying patterns include:

• Reducing complex expressions into simpler forms.
• Facilitating derivations and proofs.
• Allowing generalization to more complex situations.

Patterns are like strategies that, when identified, reveal how seemingly unique problems are part of broader, universal principles.

Expression Simplification

Expression simplification in mathematics refers to the process of rewriting an expression to make it easier to understand or solve. This can involve reducing the number of terms, combining like terms, or using mathematical notation like sums to express a sequence succinctly.
With summation notation, you can represent lengthy expressions neatly with symbols. For example, in problem (a) of the exercise, the original long-form expression \(x_{1}(x_{1}-1) + 2x_{2}(x_{2}-1) + 3x_{3}(x_{3}-1)\) is simplified to \(\sum_{n=1}^{3} n \times x_n (x_n - 1)\), condensing three terms into a concise summation.
Simplifying expressions helps to:

• Enhance clarity and understanding.
• Reduce computational difficulty.
• Allow for easier manipulation in further calculations.

Through simplification, expressions become less intimidating, and the mathematical logic underneath becomes more apparent. This skill is fundamental in both academic and real-world problem-solving contexts.