Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum of the finite arithmetic sequence. Sum of the first 100 positive integers

Step-by-Step Solution

Answer

The sum of the first 100 positive integers is 5050.

Step 1: Understanding the Formula

An arithmetic sequence is a sequence of numbers where the difference between any two successive members is a constant. A finite arithmetic sequence is an arithmetic sequence with a fixed number of terms. The formula to find the sum (S) of an arithmetic sequence is given by \(S = n/2 * (a + l)\), where 'n' is the number of terms, 'a' is the first term, and 'l' is the last term. For this problem, we know 'n=100', 'a=1', 'l=100'.

Step 2: Substitute into the formula

Substituting these values into the formula, we get \(S = 100/2 * (1 + 100) = 50 * 101\).

Step 3: Calculate the Sum

Finally, after simplifying, we find that \(S = 5050\).

Key Concepts

Sum of Arithmetic Sequence

An arithmetic sequence involves numbers arranged in a specific order where each number increases by a constant difference. Suppose you have a sequence like 1, 2, 3, ..., 100, and you want to find the total value when all these numbers are added together. This total value is called the "sum of the arithmetic sequence." The formula to determine this can be expressed as: \[ S = \frac{n}{2} \times (a + l) \]Here, "\(S\)" represents the sum, "\(n\)" is the number of terms in the sequence, "\(a\)" is the first term, and "\(l\)" is the last term. This concise formula allows you to easily calculate the total sum without adding each number individually.

Arithmetic Sequence Formula

The arithmetic sequence formula is an essential tool when working with sequences of numbers. It provides a structured mathematical way to find either a particular term in the sequence or the sum of all terms. ### The Formula ExplainedThe formula for the sum of an arithmetic sequence is:\[ S = \frac{n}{2} \times (a + l) \]**Components of the Formula:**

• "\(n\)": Denotes the total number of terms within the sequence.
• "\(a\)": The first term which starts the sequence.
• "\(l\)": The last term in the sequence.
• "\(S\)": The sum of the entire sequence of terms.

This formula can be applied to any arithmetic sequence you encounter, making calculations streamlined and efficient.

Sequence of Numbers

A sequence of numbers forms the backbone of arithmetic sequences. It is essentially a list of numbers following a particular pattern. ### Characteristics of a Sequence

• Each number in a sequence is termed as a "term."
• In an arithmetic sequence, every term increases by a constant, known as the "common difference."
• The sequence holds a fixed structure, where starting with the first term, you add the constant difference to find subsequent terms.

Understanding how sequences are constructed helps you in mastering how arithmetic sequences work, as it’s all about the pattern these terms follow.

Positive Integers Sum

When dealing with arithmetic sequences composed of positive integers, it's also crucial to know how to sum these efficiently. The sum of these integers in sequence can be determined easily by using the arithmetic sum formula. ### The Role of Positive Integers

• Positive integers are numbers greater than zero, like 1, 2, 3, etc.
• An arithmetic sequence of positive integers is straightforward, often starting from the number one and increasing by one each time.
• They offer a simple, clear path to understanding how arithmetic progresses numerically.

The simplicity makes them superb examples for students learning arithmetic sequences and their sums. Using our sequence formula makes the summing process quick and reliable.