Problem 49 - Algebra and Trigonometry

Question

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+\frac{14}{14+1} $$

Step-by-Step Solution

Answer

In summation notation, this sum can be expressed as \[ \sum_{i=1}^{14} \frac{i}{i+1} \]

Step 1: Understand the pattern in the sequence

First, notice the structure of the sequence. You will see that the numerator is steadily increasing by 1 at each term, beginning with 1. Likewise, the denominator is also increasing in each term, but it starts at 2. Therefore you can denote the numerator and denominator as i and i+1 respectively if you consider i as your index of summation which starts at 1.

Step 2: Express using summation notation

Next, use the sigma symbol (Σ) to express the above sequence in summation notation. As the problem statement indicates, use 1 as the lower limit of summation and i for the index of summation. Following this, the sequence can be written as follows: \[\sum_{i=1}^{14} \frac{i}{i+1}\]Here, i denotes the index of summation, and it runs from 1 to 14. The term to the right of the sigma symbol represents the pattern observed in the sequence.

Key Concepts

Sequence Patterns

When working with series and sequences, identifying the pattern is the crucial first step. This involves examining each term and discovering how it changes from one to the next. In the series given in the exercise, each term is of the form \( \frac{i}{i+1} \), where \( i \) represents both the numerator and the denominator with a small increment to reach the next denominator value.

Sequence patterns often follow a logical mathematical relationship, which could be arithmetic, geometric, or another identifiable pattern. In arithmetic sequences, the difference between consecutive terms remains constant. Similarly, patterns can also involve more complex relationships like those seen in geometric sequences or fractional changes as seen in this exercise.

• Observe: Look at each term.
• Identify: Spot the similarities.
• Define: Write down how each element contributes to the pattern.

Understanding the sequence pattern allows us to express it efficiently using summation notation.

Indices of Summation

The index of summation is a variable used to represent the position of terms within a sequence. In summation notation, this index often plays a pivotal role in defining the terms of the series. In our exercise, the index \( i \) is used for both the numerator and the core part of the denominator.

The lower limit of summation starts at \( i = 1 \), marking the beginning of the sequence, and the upper limit gives us the point at which the summation stops, here shown as \( i = 14 \). Index variables are arbitrary; they are placeholders that help in the formation of the terms. As long as the relationship remains consistent, the variable itself can be renamed without affecting the outcome. Some key points to remember:

• The index helps pinpoint each term in the sequence.
• Lower limit marks where the summation starts.
• Upper limit indicates where the summation ends.

Using indices effectively simplifies the process of writing and understanding series.

Sigma Notation

Sigma notation is a compact way of representing long sums with a defined pattern. The symbol \( \Sigma \) (Greek letter sigma) is used to indicate the summation of terms. In this notation, the limits of summation and the expression for the terms are critical components. For the given problem, the sigma notation is \[ \sum_{i=1}^{14} \frac{i}{i+1} \].

In sigma notation, right below the \( \Sigma \) symbol, the index of summation, such as \( i = 1 \), is defined to indicate the starting point. Above the \( \Sigma \), the number 14 specifies the ending point for such indices. Between these limits, the expression \( \frac{i}{i+1} \) represents each term that needs to be summed up.
The simplicity of sigma notation is that it allows capturing complex sequences succinctly:

• Helps avoid writing out long sums.
• Clearly displays start and end of summation.
• Highlights the repetitive nature of sequence terms.

Sigma notation thus streamlines computations, conveying much information in a simple, compact format.

Algebraic Expressions

Algebraic expressions describe numbers using variables and mathematical operations. In the context of summation, they define the form of each term in the sequence clearly, making the computation feasible. In this exercise, the expression for each term is \( \frac{i}{i+1} \). Here, this expression dictates the numerator and denominator based on the summation index \( i \).

Algebraic expressions can include:

• Constants
• Variables
• Arithmetic operations like addition, subtraction, multiplication, and division

The beauty of algebraic expressions lies in their ability to generalize patterns using one simple statement which helps to apply mathematical reasoning in various situations. Understanding how to form and manipulate these expressions is crucial not just for solving equations, but in representing datasets and mathematical ideas efficiently.