Problem 19 - Algebra 2

Question

For each sum, find the number of terms, the first term, and the last term. Then evaluate the series. $$ \sum_{n=1}^{5}(2 n-1) $$

Step-by-Step Solution

Answer

The series \(\sum_{n=1}^{5}(2n-1)\) has 5 terms. The first term is 1 and the last term is 9. The sum of the series is 25.

Step 1: Determine the number of terms

The series is given as \(\sum_{n=1}^{5}(2n-1)\). In any series of the form \(\sum_{n=a}^{b}(f(n))\), where \(a\) and \(b\) are the lower and upper limits respectively, the number of terms \(N\) can be calculated by the formula \(N = b - a + 1\). Here, \(a=1\) and \(b=5\), so \(N = 5 - 1 + 1 = 5\).

Step 2: Identify the first term

The series is defined by the function \(f(n) = 2n-1\). We can find the first term \(a_1\) by substituting \(n = 1\) into the function, yielding \(a_1 = 2(1) - 1 = 1\). So, the first term is 1.

Step 3: Identify the last term

Similarly, we can find the last term \(a_N\) by substituting \(n = N = 5\) into the function, yielding \(a_N = 2(5) - 1 = 9\). So, the last term is 9.

Step 4: Evaluate the series

The sum of an arithmetic series can be found using the formula \(S = N/2 * (a_1 + a_N)\), where \(S\) is the sum, \(N\) is the number of terms, \(a_1\) is the first term and \(a_N\) is the last term. Substituting the values from steps 1 through 3 into this formula produces \(S = 5/2 * (1 + 9) = 5/2 * 10 = 25\)

Key Concepts

Algebraic Formulas

Algebraic formulas are essential tools that help simplify the solving process for various mathematical problems, including series. Arithmetic series, like the one given \(\sum_{n=1}^{5}(2n-1)\), rely on formulas to easily determine their characteristics. In this series, each term follows a pattern defined by the algebraic expression \(f(n) = 2n - 1\). This expression provides a simple way to find any term within the series by substituting the value of \(n\).
For instance, to find the first term, substitute \(n=1\) into the function, resulting in 1. Similarly, by substituting \(n=5\), we find the last term is 9. Using such algebraic expressions makes it efficient to analyze the elements within the series and understand the relationship between consecutive terms.

Series Evaluation

Series evaluation involves finding the sum of all terms in a sequence. For an arithmetic series, there's a specific formula that simplifies this process: \[S = \frac{N}{2} \times (a_1 + a_N)\]. Here, \(S\) represents the sum of the series, \(N\) indicates the number of terms, \(a_1\) is the first term, and \(a_N\) is the last term.
Using the example of the series \(\sum_{n=1}^{5}(2n-1)=S\), we first determine \(N\), \(a_1\), and \(a_N\) using earlier steps. In this case, we have 5 terms, starting with 1 and ending with 9. Plugging these values into our formula gives us: \[S = \frac{5}{2} \times (1 + 9) \]. Simplifying this, we find \(S = \frac{5}{2} \times 10 = 25\). This calculation efficiently evaluates the series sum without manually adding each term.

Number of Terms

Determining the number of terms in a series is a crucial step when evaluating it. The general formula to find the number of terms \(N\) in a series \(\sum_{n=a}^{b}(f(n))\) is \(N = b - a + 1\). Here, \(a\) is the starting index and \(b\) is the ending index, which helps us calculate how many terms exist between these two points inclusive.
In our specific series \(\sum_{n=1}^{5}(2n-1)\), we apply the formula: \(N = 5 - 1 + 1 = 5\). This tells us there are 5 terms. It's important to determine \(N\) accurately as it directly impacts the evaluation of the series sum using arithmetic series formulas. Knowing the number of terms aids in understanding the series structure and ensures all calculations consider each element effectively.