Problem 33 - Algebra for College Students

Question

Write each series in summation notation. Use the index is and let i begin at I in each summation. $$w_{1}+w_{2}+w_{3}+\dots+w_{n}$$

Step-by-Step Solution

Answer

\( \displaystyle\sum_{i=1}^{n} w_{i} \)

Step 1 - Identify the terms of the series

Notice that the terms of the series are given as \( w_{1}, w_{2}, w_{3}, \dots , w_{n} \). Each term is indexed by i, starting from 1 to n.

Step 2 - Understand summation notation

Summation notation is a concise way to represent a series using the Greek letter \( \Sigma \). The general format is \[ \displaystyle\sum_{i=starting \ value}^{ending \ value} term \].

Step 3 - Apply summation notation

Write the given series in summation notation: \[ \displaystyle\sum_{i=1}^{n} w_{i} \].

Key Concepts

series representation

In mathematics, a series is the sum of the terms of a sequence. When you have a list of numbers, like 1, 2, 3, 4... adding them together forms a series. This concept is powerful as it allows us to work with potentially infinite sums in a manageable way.

Series representation is crucial because it simplifies notation and calculations. Instead of writing out every term, you can use a compact form. For example, the series for the numbers 1 through n can be written as 1 + 2 + 3 + ... + n. Using a shorthand like summation notation (Σ) saves time and helps in understanding and analyzing patterns within the series.

index notation

Index notation is a way of writing the terms of a series using indices. Indices are typically represented by letters like i, j, or k to denote their position in the series. For the given series, the terms are written as \(w_{1}, w_{2}, w_{3}, \text{...}, w_{n}\), where the subscript (index) indicates the term's position.

This system allows mathematicians to generalize and manipulate terms without relying on specific values. For instance, \(w_{i}\) refers to any term in the series, allowing us to sum all terms succinctly. When writing the series starting with i and using summation notation, it becomes \(\text{\textbackslash sum_{i=1}^{n} w_{i}}\) , where i runs from 1 to n.

Greek letters in math

Greek letters play a significant role in mathematical notation. They are used to represent constants, variables, functions, and summation symbols. One of the most common Greek letters in mathematics is Σ (Sigma), used to indicate summation.

For instance, \(\text{\textbackslash sum}\), the symbol for summation, compactly represents the sum of a set of terms. In the given problem, \(\text{\textbackslash sum_{i=1}^{n} w_{i}}\) denotes summing the \(i^{th}\) term of the series from i = 1 to n. Recognizing and understanding these symbols is vital for reading and writing mathematical expressions efficiently. Other Greek letters like Δ (Delta), π (Pi), and α (Alpha) also frequently appear in various mathematical contexts.