Problem 1 - Discrete Mathematics with Applications

Question

Evaluate each sum. $$\sum_{i=1}^{6} i$$

Step-by-Step Solution

Answer

The sum of the arithmetic sequence \(1 + 2 + 3 + 4 + 5 + 6\) can be evaluated using the formula \(S_n = \frac{n (a_1 + a_n)}{2}\), where \(n = 6\), \(a_1 = 1\), and \(a_n = 6\). Substituting these values and simplifying the expression, we find the sum to be \(S_6 = \frac{6 (1 + 6)}{2} = \frac{6 (7)}{2} = \frac{42}{2} = 21\).

Step 1: Recognize the arithmetic sequence

The problem can be rephrased as "Find the sum of the arithmetic sequence with the first term being 1 and the last term being 6." The series is 1 + 2 + 3 + 4 + 5 + 6, which is an arithmetic sequence with a common difference of 1.

Step 2: Use the formula for the sum of an arithmetic sequence

To find the sum of an arithmetic sequence, we can use the following formula: $$ S_n = \frac{n (a_1 + a_n)}{2} $$ Here, \(S_n\) is the sum of the sequence, \(n\) is the number of terms in the sequence, \(a_1\) is the first term, and \(a_n\) is the last term.

Step 3: Substitute the given values

In this problem, \(n = 6\), \(a_1 = 1\), and \(a_n = 6\). Substitute these values into the formula: $$ S_6 = \frac{6 (1 + 6)}{2} $$

Step 4: Simplify and calculate the result

Now simplify the expression and calculate the sum of the sequence: $$ S_6 = \frac{6 (7)}{2} = \frac{42}{2} = 21 $$ Hence, the sum of the specified arithmetic sequence is 21.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is an ordered list of numbers where each term after the first is found by adding a constant amount, known as the common difference, to the previous term. The terms of an arithmetic sequence can be denoted as \(a_1, a_2, a_3, ...\), where \(a_1\) is the first term, and the common difference is given by \(a_{n} - a_{n-1}\).

For example, in the given exercise, the sequence \(1, 2, 3, 4, 5, 6\) is an arithmetic sequence with \(a_1 = 1\) and the common difference \(d = 1\), which means each subsequent term increases by 1. Understanding the structure of an arithmetic sequence is essential for finding its sum, as it allows us to use a formula that simplifies the process.

Summation Notation

Summation notation, represented by the Greek letter sigma \( \Sigma \), provides a concise way to express the sum of a series of terms. It's used to add up a sequence of numbers over a particular index, which is often represented as \(i\), \(j\), or another variable. When applying summation notation:

• The variable below the \( \Sigma \) symbol indicates the starting index of the sum.
• The number above \( \Sigma \) symbol denotes the ending index.
• The expression to the right of \( \Sigma \) describes the term to be added, which can be a function of the index.

In the previous exercise, \( \sum_{i=1}^{6} i \) means the sum of all integers from 1 to 6. Summation notation is crucial as it enables us to write long sums in a compact, understandable form and work with sequence and series in an organized manner.

Sequence and Series

While a sequence is a set of numbers in a specific order, a series is the sum of the terms of a sequence. Arithmetic series, geometric series, and more complex series are all topics in this branch of mathematics. In essence, a series takes the individual elements of a sequence and combines them through addition.

For arithmetic sequences in particular, the arithmetic series can be summed up by a formula derived from the sequence's uniformity. In the case of the exercise, the sequence and series concepts converge in the calculation of the sum of the first six positive integers. Understanding these concepts allows students to handle problems involving regular patterns and summations, which are prevalent in various mathematical and real-life applications.