Problem 50 - Algebra and Trigonometry

Question

Expand each sum. \(\sum_{k=1}^{n}(2 k+1)\)

Step-by-Step Solution

Answer

The expanded sum is \(3 + 5 + 7 + \ldots + (2n+1)\).

Step 1: Identify the general term of the sum

Recognize that the general term in the sum is given by \(2k+1\), where \(k\) ranges from \(1\) to \(n\).

Step 2: Expand the sum

Write out the sum term by term. When \(k = 1\), the term is \(2(1) + 1 = 3\); when \(k = 2\), the term is \(2(2) + 1 = 5\); and so on up to \(k = n\). Thus, the expanded sum is \( \sum_{k=1}^{n}(2k+1) = (2 \times 1 + 1) + (2 \times 2 + 1) + (2 \times 3 + 1) + \ldots + (2 \times n + 1) = 3+5+7+ \ldots + (2n+1).\)

Step 3: Combine the terms in a single equation

Combine all the expanded terms for clarity: \( \sum_{k=1}^{n}(2k+1) = 3 + 5 + 7 + \ldots + (2n+1).\)

Key Concepts

Summation Notation

Summation notation is a concise way of expressing the sum of a sequence of numbers. The symbol for summation is the Greek letter sigma, \(\sum\). Below this symbol, we write the index variable (commonly \(k\)) and its range. For instance, in the sum \(\sum_{k=1}^{n}(2k+1)\), \(\(k=1\) to \(k=n\)\) specifies that \(k\) starts at 1 and increments by 1 until it reaches \(n\). This notation is very useful as it simplifies the expression of sums, especially for long series where writing out each term manually would be impractical.

Series Expansion

Series expansion refers to the process of writing out each term of a series explicitly. In the exercise, we expanded the summation \(\sum_{k=1}^{n}(2k+1)\) by substituting \(k\) with each integer value from 1 to \(n\). The first term is derived as \(2(1)+1 = 3\), the second term as \(2(2)+1 = 5\), and so on. The sequence continues until the general term for \(k = n\) is reached, which is \(2n+1\). Thus, the expanded series is \(3 + 5 + 7 + \ldots + (2n+1)\).

General Term Identification

Identifying the general term of a sequence is crucial for understanding and expanding series. The general term defines each element of the sequence based on the index variable \(k\). In the sum given in the exercise, the general term is \(2k+1\). To identify this, observe how each term in the series is formed. As \(k\) increases by 1, the term \(2k+1\) increases accordingly because it directly depends on \(k\). This general term helps in not only expanding but also in analyzing and summarizing the behavior of the entire series.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. The general term \(2k+1\) in the summation is an example of an algebraic expression. Here, \(2k\) represents the product of 2 and the variable \(k\), while \(1\) is a constant term added to the product. Understanding how to manipulate algebraic expressions is essential for expanding, simplifying, and solving mathematical problems. When expanding the sum \(\sum_{k=1}^{n}(2k+1)\), each term is a result of substituting \(k\) into the algebraic expression \(2k+1\).