Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Express the sum in terms of summation notation. (Answers are not unique.) $$3+\frac{3}{5}+\frac{3}{25}+\frac{3}{123}+\frac{3}{625}$$

Step-by-Step Solution

Answer

\( \sum_{n=0}^{4} 3 \left(\frac{1}{5}\right)^n \)

Step 1: Identify the Pattern

Observe the sequence of numbers: \(3, \frac{3}{5}, \frac{3}{25}, \frac{3}{125}, \frac{3}{625}\). Notice that each term can be expressed as \(3\times\left(\frac{1}{5}\right)^n\), where \(n\) represents the position of the term starting from 0.

Step 2: Write the General Term

The general term of the sequence can be written as \(3 \left(\frac{1}{5}\right)^n\), where \(n=0,1,2,3,4\). This represents each term in the sequence.

Step 3: Express Using Summation Notation

Using the general term identified in Step 2, express the sum using summation notation: \[ \sum_{n=0}^{4} 3 \left(\frac{1}{5}\right)^n \]This notation compactly represents the sum of the sequence terms from \(n=0\) to \(n=4\).

Key Concepts

Understanding Geometric Series

A geometric series is a type of sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In our case, the sequence provided in the exercise represents a geometric series. Each term is the previous term multiplied by \( \frac{1}{5} \), which is our common ratio.
To define a geometric series, it's essential to identify two elements:

• The initial term, which is 3 in this series.
• The common ratio, which is \( \frac{1}{5} \).

Geometric series are prevalent in various mathematical applications, including calculating compounded interest or modeling exponential growth. Understanding how each term contributes to the series helps solve complex summations easily. Recognizing this underlying structure allows you to express sums efficiently like we've done using summation notation.

Identifying the Sequence Pattern

Identifying patterns in sequences is key to understanding and solving many mathematical problems. For the given series, pattern recognition involved noticing how each term was related to the one before it.
Upon examining the sequence \( 3, \frac{3}{5}, \frac{3}{25}, \frac{3}{125}, \frac{3}{625} \), we see each term is derived by multiplying the previous term by \( \frac{1}{5} \).
More generally, a sequence pattern helps to:

• Find the general rule that defines the sequence.
• Predict further terms without direct calculation.
• Simplify complex problems by recognizing simpler repeated elements.

Sequence patterns simplify our mathematical efforts by creating predictability and enabling us to model data or solve summation problems more systematically.

Formulating the General Term

The general term of a sequence serves as a blueprint to describe any term in the sequence, based on its position. For our sequence, the general term is given by the expression \( 3 \left( \frac{1}{5} \right)^n \).
In this formula:

• The number 3 represents the initial term of the sequence.
• \( \left( \frac{1}{5} \right)^n \) indicates how each subsequent term is a power of the common ratio \( \frac{1}{5} \).
• \( n \) denotes the term's position, starting from 0.

Formulating the general term is critical to expressing and solving problems efficiently, particularly when using mathematical notation such as summation. It enables mathematicians to express potentially infinite series compactly and work with them algebraically.

Utilizing Mathematical Notation

Mathematical notation provides a symbolic shorthand to convey complex ideas succinctly. In this exercise, we use summation notation to succinctly express the addition of sequence terms.
The expression \( \sum_{n=0}^{4} 3 \left( \frac{1}{5} \right)^n \) communicates the sum of the series from the first to the fifth term:

• The Greek letter \( \Sigma \) represents 'sum'.
• The subscript \( n=0 \) and superscript 4 in the notation indicate that \( n \) starts at 0 and ends at 4.
• The function \( 3 \left( \frac{1}{5} \right)^n \) describes each term in the summation.

Mathematical notation transforms lengthy explanations into a format that is easy to understand and manipulate mathematically. It allows us to work with the general concept effectively, making calculations or further transformations easier to handle in broader mathematical contexts.