Problem 26 - Algebra for College Students

Question

Find the sum \(5+7+9+\cdots+137\) 4757

Step-by-Step Solution

Answer

The sum of the sequence is 4757.

Step 1: Determine the Sequence Type

First, notice that the sequence starts with 5 and the difference between consecutive terms is 2 (i.e., 5, 7, 9,...). This is an arithmetic sequence.

Step 2: Identify the First Term and Common Difference

Identify the first term, which is 5 (\(a_1 = 5\)), and the common difference, which is 2 (\(d = 2\)).

Step 3: Find the Number of Terms in the Series

The formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1) imes d\). We need to find \(n\) when \(a_n = 137\). Solve \(137 = 5 + (n-1) imes 2\).1. Simplify: \(137 = 5 + 2n - 2\).2. Combine like terms: \(137 = 3 + 2n\).3. Subtract 3 from both sides: \(134 = 2n\).4. Divide by 2: \(n = 67\).So, there are 67 terms.

Step 4: Use the Arithmetic Series Sum Formula

The formula for the sum of an arithmetic series is \(S_n = \frac{n}{2} \times (a_1 + a_n)\).Plug in the values: \(S_{67} = \frac{67}{2} imes (5 + 137)\).1. Calculate the terms inside the brackets: \(5 + 137 = 142\).2. Multiply: \(S_{67} = \frac{67}{2} imes 142\).3. Divide 67 by 2: \(33.5 \times 142\).4. Calculate: \(S_{67} = 4757\).

Step 5: Conclude the Calculation

Therefore, the sum of the arithmetic series from 5 to 137 with a common difference of 2 is 4757.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial element. It defines how much we add to each term to get the next one. This means that each consecutive term in the sequence is derived by adding the common difference to the previous term. For example, if we have the sequence starting from 5, as in our problem, and a common difference of 2, then the terms will line up as: 5, 7, 9, 11, and so on.
A helpful way to remember this is to think of the sequence as a regular pattern, rhythmically growing by the same amount each time. The mathematical representation of this is given by the formula, where if the first term is denoted by \(a_1\) and the common difference by \(d\), then the nth term \(a_n\) is expressed as \(a_n = a_1 + (n-1) \, d\).

• First term \(a_1 = 5\)
• Common Difference \(d = 2\)

Thus, identifying the common difference helps you predict any term in the sequence.

Sum of Arithmetic Series

Once you know the terms of an arithmetic sequence, you might need to find the sum of these terms. This is where the sum of an arithmetic series comes in.
The formula to find this sum is \(S_n = \frac{n}{2} \, (a_1 + a_n)\), where \(n\) is the total number of terms, \(a_1\) is the first term, and \(a_n\) is the nth term.

• In our exercise, the sum is calculated as \(S_{67} = \frac{67}{2} \, (5 + 137)\).
• Simplified, this equals \(33.5 \, \times \, 142\), leading us to the final sum of 4757.

This formula elegantly combines the first and last terms, highlighting how an arithmetic series adds up, creating a symmetry around its center.

Nth Term Formula

To navigate an arithmetic sequence, the nth term formula is like a roadmap. It shows you how to locate any term in the sequence without listing all the previous ones.
The formula for the nth term is \(a_n = a_1 + (n-1) \, d\), which directly uses the first term \(a_1\) and the common difference \(d\) to compute any term \(a_n\).
For example, in our specific problem, to find the 67th term, the equation would be:

• Start with the first term: \(a_1 = 5\)
• Use the common difference: \(d = 2\)
• Calculate: \(a_{67} = 5 + 66 \, \times \, 2 = 137\)

Thus, this formula helps in finding the number of terms necessary for determining the sum accurately.

Sequence Calculation

Calculating a sequence involves understanding both its structure and the relations between its terms. In the context of arithmetic sequences, this is largely guided by the identification of the first term, the common difference, and applying this framework to determine either specific terms or sums.
Here’s how you could systematically approach this:

• Identify the first term (e.g., \(a_1 = 5\)).
• Determine the common difference (e.g., \(d = 2\)).
• Use the nth term formula to explore different terms of interest.
• Calculate the total number of terms \(n\) when a specific term \(a_n\) is given (e.g., \(a_{67} = 137\)).
• Finally, apply the sum formula to compute the total sum.

Mastering these steps facilitates sequence calculations by turning them into an orderly process of discovery and verification, ensuring precision and confidence in your math work.