Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find the partial sum. $$\sum_{n=1}^{20}(2 n+1)$$

Step-by-Step Solution

Answer

The partial sum of the series is 440.

Step 1: Identify the Rule for the Series

The given series is \(\sum_{n=1}^{20}(2n+1)\). The nth term rule for this arithmetic series is defined by \(a_n = 2n + 1\). This means that each term of the series increases by 2 from the previous term.

Step 2: Use the Formula for the Sum of an Arithmetic Series

The sum \(S\) of the first n terms of an arithmetic series can be found using the formula \(S = n/2 * (a_1 + a_n)\), where \(a_1\) is the first term, \(a_n\) is the nth term, and n is the number of terms. So, using this formula for our series, we substitute \(n = 20\), \(a_1 = a_1 = 2*1 + 1 = 3\), and \(a_n = a_20 = 2*20 + 1 = 41\).

Step 3: Calculate the Partial Sum

Substitute the values into the formula to get \(S = 20/2 * (3 + 41) = 10 * 44 = 440\).

Key Concepts

partial sum

In arithmetic series problems, the term 'partial sum' refers to the sum of a certain number of terms in the sequence, rather than the entire sequence which might go on indefinitely. For our series, \(\sum_{n=1}^{20}(2n+1)\), the partial sum includes the first 20 terms. Calculating this sum helps us to understand the behavior and the total value within this specific range of the sequence.

• The partial sum for our exercise focuses only on the terms from the start (\(n=1\)) to the 20th term (\(n=20\)).
• Adding up each term based on the nth term rule (\(a_n = 2n + 1\)) gives us an insight into the sum within that segment of the sequence.

By finding the partial sum, we can analyze specific portions of an arithmetic series without needing to deal with the entire set of terms, which can be particularly useful in identifying patterns and ensuring calculations are manageable.

nth term

The 'nth term' of an arithmetic series is a formula that defines each term in the sequence based on its position. In our example, the rule for the nth term is given by:
\(a_n = 2n + 1\).
This means every term in the series can be calculated by plugging in the position number \(n\) into this formula.

• The first term (\(a_1\)) is \(2(1) + 1 = 3\).
• The second term (\(a_2\)) is \(2(2) + 1 = 5\), and this pattern continues.
• The nth term rule allows you to determine any term in the sequence without having to add up all preceding terms.

Understanding the nth term is crucial because it provides a direct relationship between the position of a term and its value in the series. This makes it straightforward to find any particular term, especially the last term in a finite sequence, as needed for calculating partial sums.

sum formula

The sum formula for arithmetic series simplifies finding the total of a sequence of numbers. Specifically, the sum \(S\) of the first \(n\) terms of an arithmetic series can be calculated using the formula:
\[S = \frac{n}{2} \times (a_1 + a_n)\]Here, \(n\) stands for the number of terms, \(a_1\) is the first term, and \(a_n\) is the nth term.

• In our series, \(n = 20\), \(a_1 = 3\), and \(a_{20} = 41\).
• Plugging these into the formula gives us \(S = \frac{20}{2} \times (3 + 41)\).
• This simplifies to \(10 \times 44\), resulting in a total sum of \(440\).

This formula is powerful because it quickly calculates the sum of a series without having to add each term manually. By using key values derived from the series, it offers efficiency and accuracy, especially for sequences with a large number of terms.

graphing utility

A graphing utility is a tool, often software or a calculator, used to visualize mathematical equations and sequences. When working with series such as an arithmetic series, a graphing utility can be incredibly helpful.

• It allows you to plot the series and observe the growth pattern.
• You can visually confirm the increasing nature of the series, given the consistent incremental step by which each term increases.
• Graphing utilities often include features for calculating sums, providing an alternate method to verify manual calculations.

In this exercise, you could use a graphing utility to enter the function \(a_n = 2n + 1\) and graphically see the range from \(n=1\) to \(n=20\). By observing the graphical representation, you gain a deeper understanding and a more intuitive grasp of how the sum is built up, complementing the numerical methods we perform manually.