Problem 59 - Algebra and Trigonometry

Question

\(55-60\) . Write the sum without using sigma notation. $$ \sum_{k=3}^{100} X^{k} $$

Step-by-Step Solution

Answer

The sum without sigma notation is \( X^{3} + X^{4} + X^{5} + \ldots + X^{100} \).

Step 1: Understand Sigma Notation

The sigma notation \( \sum_{k=3}^{100} X^{k} \) represents the sum of the terms where the exponent of \( X \) starts at 3 and goes up to 100. This means you need to write out the series of terms starting from \( X^3 \) up to \( X^{100} \).

Step 2: Write the First Few Terms

Start by writing out the first few terms of the series: \( X^{3} + X^{4} + X^{5} + X^{6} + \ldots \). This gives an idea of how the series is structured.

Step 3: Continue the Pattern

Continue the pattern by writing out all terms from the start to the end of the series. Increase the exponent by 1 for each subsequent term, at least covering a portion to show continuity: \( X^{3} + X^{4} + X^{5} + \ldots + X^{99} + X^{100} \).

Step 4: Write the Sum Without Sigma

Finally, write the entire sum without using sigma notation: \( X^{3} + X^{4} + X^{5} + X^{6} + \ldots + X^{98} + X^{99} + X^{100} \). Ensure that all terms are correctly represented.

Key Concepts

Series Expansion

In mathematics, a series expansion allows us to express a summed series of terms in a more accessible form. Consider the problem \( \sum_{k=3}^{100} X^{k} \), where we need to expand the series without using sigma notation.
This involves writing out each term starting from \(X^3\) and continuing to \(X^{100}\).
The idea is to replace the sigma notation, \( \sum \), with all terms it represents in its expanded form:

• The series starts at \(X^3\).
• Each subsequent term increases the exponent by 1: \(X^3, X^4, X^5, \ldots\).
• The series concludes with \(X^{100}\).

By following this pattern, the series becomes a straightforward, visible list of terms. This makes understanding and manipulation easier, providing a clearer view of the complete range of values involved.

Exponential Terms

Exponential terms are expressions where a constant base, such as \(X\), is raised to the power of an incrementing variable. In our exercise, the base \(X\) remains the same, but the exponent changes from 3 to 100.
The general form of an exponential term in this context is expressed as:

• \(X^k\), where \(k\) is the exponent.

Here are a few characteristics to help identify and work with exponential terms:

• As \(k\) increases, the value of \(X^k\) also increases, unless \(0 < X < 1\), which causes the opposite to happen.
• Exponential terms grow rapidly, especially when the base \(X\) is larger than one.
• In our series, each term \(X^k\) is distinct due to its unique exponent.

Understanding these properties aids in managing such terms in broader algebraic expressions or series expansions.

Algebraic Expressions

In algebra, expressions combine variables, constants, and various operations. The series \(X^3 + X^4 + X^5 + \ldots + X^{100}\) is an example of an algebraic expression structured as a sum of exponential terms.
Algebraic expressions can be complex, involving multiple different operations. In our case, we're working strictly with addition and exponential operations:

• Each term follows a simple power law, increasing consecutively.
• Sum these exponential terms to form a more significant algebraic expression.

Simplifying or understanding such a sequence requires recognizing patterns and homogeneity in terms. For instance:

• Consistency in base \(X\).
• Identifying the progression from one term to the next.

By breaking down the components and understanding how they relate, you can manage and manipulate complex algebraic expressions effectively.