Problem 56 - Precalculus with Limits

Question

In Exercises 51 - 58, find the sum of the finite arithmetic sequence. Sum of the first \( 100 \) positive odd integers

Step-by-Step Solution

Answer

The sum of the first 100 positive odd integers is 10000.

Step 1: Identify the first and last term

The first term (\(a\)) of the sequence is the first positive odd integer, which is 1. The last term of the sequence is the 100th odd number. The nth odd number can be found using the formula \[2n - 1\]. Plugging 100 for n, the 100th odd number is \[2(100) - 1 = 199\]. So, the last term (\(l\)) is 199.

Step 2: Identify the number of terms

The number of terms (\(n\)) in the sequence is given as 100.

Step 3: Use the Sum of Arithmetic sequence formula

Now, we will use the formula for the sum of an arithmetic sequence to find the answer: \[S = \frac{n}{2}(a + l)\]. Hence, substituting the values of \(n\), \(a\), and \(l\), we get \(S = \frac{100}{2}(1 + 199)\).

Step 4: Simplify the expression

Solving the equation gives us \(S = 50 \times 200 = 10000\).

Key Concepts

Sum of Finite Sequence

When seeking the sum of a finite sequence, specifically an arithmetic sequence, it's important to understand the definition first. An **arithmetic sequence** is a series of numbers in which each term increases or decreases by a constant difference from the previous term. This difference is called the *common difference*, often represented as \(d\).
In our particular problem, we are asked to find the sum of the first 100 positive odd integers. It's a classic example of a finite sequence.

To calculate the sum, we do not need to add each number manually. Instead, we take advantage of the **sum formula for arithmetic sequences**. This formula allows us to swiftly compute the sum by using:

• The number of terms (\(n\))
• The first term (\(a\))
• The last term (\(l\))

Utilizing these variables, the formula is expressed as:
\[S = \frac{n}{2}(a + l)\]
By plugging in our values into this formula, we find the desired sum efficiently without the need for tedious addition.

Odd Integers

Odd integers are numbers that cannot be evenly divided by 2. They form a sequence of numbers that increases by a consistent increment—just like even numbers but starting from one.

For any natural number \(n\), the odd integer sequence formula \(2n - 1\) helps us identify the \(n\)-th odd number in this sequence. For instance:

• When \(n = 1\), the odd integer becomes \(2(1) - 1 = 1\)
• When \(n = 2\), the odd integer becomes \(2(2) - 1 = 3\)
• And so forth, ensuring each value is odd and consistently spaced

In our exercise, we used \(n = 100\) to determine that the 100th odd integer is 199, confirming our arithmetic sequence's first and last terms.

Sum Formula

The **sum formula** for an arithmetic sequence is a powerful tool that helps simplify the process of adding sequences with a regular pattern. For an arithmetic sequence characterized by a first term \(a\), a number of terms \(n\), and a last term \(l\), the sum \(S\) is calculated by:
\[S = \frac{n}{2}(a + l)\]
This formula helps you determine the total sum of all terms in the sequence in a single, straightforward calculation.

Here's a brief breakdown of the components of this formula:

• \(n\): Total number of terms in the sequence.
• \(a\): The first term of the sequence.
• \(l\): The last term in the sequence.
• The fractional term \(\frac{n}{2}\) calculates the average number of pairs in the sequence.

In practice, like in our problem, this formula allows for quick computation when summing even a large number of neatly ordered terms, offering not just simplicity but also efficiency. By using the provided values, the sum of the first 100 odd integers is easily found to be 10,000.