Problem 57 - Algebra and Trigonometry

Question

Write the sum without using sigma notation. $$\sum_{k=3}^{100} x^{k}$$

Step-by-Step Solution

Answer

The sum is \( x^3 + x^4 + x^5 + \ldots + x^{100} \).

Step 1: Understand the Sigma Notation

The given expression \( \sum_{k=3}^{100} x^k \) tells us to sum all terms of the form \( x^k \), starting with \( k=3 \) and ending with \( k=100 \).

Step 2: Translate into Expanded Form

To write the sum without using sigma notation, we need to expand this expression. The terms of the series are \( x^3, x^4, x^5, \ldots, x^{100} \).

Step 3: List Out All Terms

Write out all the terms individually in a list:\[x^3 + x^4 + x^5 + \ldots + x^{100}\] This is the expanded form of the series.

Key Concepts

Series Expansion

When you see something like \( \sum_{k=3}^{100} x^k \), you are looking at a compact way to express a series of numbers or terms. Series expansion is the technique of expressing this compact notation into a long sum, written out term by term. Each term in the series follows a predictable pattern, given by the expression involving \( k \), the index of summation. In this case, each term is \( x^k \).

There are several reasons to expand series like this:

• To understand what each term looks like explicitly.
• To make computations or evaluations clearer when actual numbers replace the variables.
• To help visualize the progression or pattern of the series more readily.

Going from sigma notation to expanded form essentially helps by laying out each component part clearly, making any subsequent operations, understanding, or analysis straightforward.

Powers of x

The terms in the series \( x^3, x^4, x^5, \ldots, x^{100} \) involve increasing powers of a variable \( x \). Here, \( x \) is raised to successive integer powers starting from 3 up to 100. Every time you increase \( k \) by 1, the power of \( x \) increases by 1 accordingly, such as from \( x^3 \) to \( x^4 \).

Working with powers of \( x \) involves understanding a few basic concepts:

• Exponents: Indicate how many times \( x \) is multiplied by itself.
• Polynomial Growth: As the exponent becomes larger, the term grows more rapidly if \( x > 1 \).
• Visualization: Often helpful to imagine or sketch, especially when comparing large powers.

This understanding allows us to think about each term separately but also how they combine to form larger structures, such as polynomials or series.

Summation Techniques

Summation techniques are strategies for adding up sequences of terms efficiently, especially when they involve variables and extend over many terms like our example from 3 to 100. The sigma notation \( \sum \) is just a starting point to set up the sum to engage various techniques:

• Direct Summation: If possible, simply writing out the terms and adding them is one strategy, especially useful when terms are straightforward.
• Using Formulas: For some series, there are known formulas to compute the sum directly without listing all terms, but in our case with powers of \( x \), such a formula isn't typically straightforward.
• Approximation: In more advanced contexts, approximate techniques can speed up computation when exact terms are known to be cumbersome.

Understanding these basic approaches helps in deciding how best to tackle a summation problem, depending on its specific constraints and requirements. This choice of technique can significantly affect both the time and complexity involved in finding the solution.