Problem 48 - Precalculus: Mathematical for Calculus

Question

Use a graphing calculator to evaluate the sum. $$\sum_{k=1}^{100}(3 k+4)$$

Step-by-Step Solution

Answer

The sum is 15550.

Step 1: Understand the Problem

We need to evaluate the sum of the sequence given by the expression \(3k + 4\) where \(k\) starts at 1 and ends at 100. This can be represented as \(\sum_{k=1}^{100} (3k + 4)\).

Step 2: Recognize the Sequence Structure

The sequence \((3k + 4)\) is an arithmetic sequence where each term is created by inserting values of \(k\) from 1 to 100 into the expression. The starting term is \(3(1) + 4 = 7\) and the last term is \(3(100) + 4 = 304\).

Step 3: Use the Sum Formula for Arithmetic Sequence

The sum of an arithmetic sequence can be calculated using the formula \( S_n = \frac{n}{2}(a + l) \), where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term. Here, \(n=100\), \(a=7\), and \(l=304\).

Step 4: Calculate the Sum

Substitute the known values into the formula: \[ S_{100} = \frac{100}{2}(7 + 304) \]. Simplifying these results in \[ S_{100} = 50 \times 311 = 15550 \]. This gives us the sum of the sequence.

Key Concepts

Sum of Arithmetic Sequence

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. In this exercise, the expression \(3k+4\) creates such a sequence. Let's find out how we calculate its sum.

The formula to determine the sum \(S_n\) of an arithmetic sequence is:

• \(S_n = \frac{n}{2}(a + l)\)

Where:

• \(n\) is the number of terms in the sequence.
• \(a\) is the first term.
• \(l\) is the last term.

In our example:

• \(a = 7\), as we calculated from the first term \(3(1) + 4\).
• \(l = 304\), obtained from the last term \(3(100) + 4\).
• \(n = 100\), because \(k\) ranges from 1 to 100.

Now, plug these values into the formula:

• \[ S_{100} = \frac{100}{2}(7 + 304) = 50 \times 311 = 15550 \]

The sum of the arithmetic sequence from \(k = 1\) to \(k = 100\) is 15550.

Graphing Calculator

A graphing calculator is a powerful tool for evaluating sequences and their sums. These devices not only execute arithmetic operations but also visualize sequences, making them vital in understanding mathematical concepts.

When using a graphing calculator to evaluate the sum of an arithmetic sequence like \(\sum_{k=1}^{100}(3 k+4)\), follow these steps:

• Enter the sequence expression \(3k + 4\) into the sequence mode or summation function of the calculator.
• Set your sequence boundaries starting at \(k = 1\) and ending at \(k = 100\).
• Run the calculation to find the resulting sum.

Most modern graphing calculators can perform this entire operation with a few keystrokes, providing both the exact sum and, in some models, a visual representation of the sequence.

Sequences and Series

Sequences and series are foundational concepts in mathematics. A sequence is an ordered list of numbers, and a series is the sum of the terms of a sequence.

An arithmetic sequence, specifically, has the property of an equal difference between consecutive terms. This difference is called the 'common difference'. For instance, in the sequence \(3k + 4\), the common difference is 3, as each term increases by 3 when you move to the next term by incrementing \(k\) by 1.

The sum of a series is calculated up to a specified number of terms, using formulas that simplify lengthy additions. By viewing a sum as a series, students can understand how different parts of mathematics are interconnected, such as how series relate to functions and calculus.

Recognizing patterns, such as arithmetic sequences in the context of series, empowers students. These patterns offer a structured approach and enhance problem-solving skills.