Problem 35 - Algebra for College Students

Question

Find the sum of the first 50 terms of the arithmetic sequence \(5,7,9,11,13, \ldots\). 2700

Step-by-Step Solution

Answer

The sum is 2700.

Step 1: Identify the terms of the sequence

The given sequence is an arithmetic sequence where each term increases by the same amount from the previous term. The first term (\(a\)) is 5, and the common difference (\(d\)) is found by subtracting the first term from the second one: \(d = 7 - 5 = 2\).

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

The sum of the first \(n\) terms of an arithmetic sequence can be calculated using the formula: \[ S_n = \frac{n}{2} \times (2a + (n-1) d) \] where \(n\) is the number of terms, \(a\) is the first term, and \(d\) is the common difference.

Step 3: Plug in the known values

Substitute the known values into the sum formula: \(n = 50\), \(a = 5\), and \(d = 2\). Then calculate: \[ S_{50} = \frac{50}{2} \times (2 \times 5 + (50-1) \times 2) \]

Step 4: Perform the calculations

Calculate the expression: \[ S_{50} = 25 \times (10 + 49 \times 2) \] Simplify inside the parentheses first: \(10 + 98 = 108\). Then calculate: \(25 \times 108 = 2700\).

Step 5: Conclude the result

The sum of the first 50 terms of the given arithmetic sequence is 2700.

Key Concepts

Sum of Arithmetic Series

An arithmetic sequence is a series of numbers in which the difference between any two successive terms is constant. This constant difference is referred to as the 'common difference'. The sum of such a sequence is known as the 'sum of an arithmetic series'. To find this sum, we employ a powerful formula:

• First, identify the number of terms, denoted as \( n \).
• Determine the initial term, \( a \).
• Calculate the common difference \( d \).

The formula is provided by:\[S_n = \frac{n}{2} \times (2a + (n-1) d)\]This formula operates by essentially taking the average of the first and last term of the sequence, and then multiplying by the number of terms. It efficiently encapsulates the regular spacing of the numbers in the sequence. For our specific problem, with \( a = 5 \), \( n = 50 \), and \( d = 2 \), the sum turns out to be 2700. This formula is incredibly helpful for quickly summing long sequences without manually adding each term.

Common Difference

The common difference in an arithmetic sequence is the consistent amount each term varies from the previous one. This difference is fundamentally what defines an arithmetic sequence. To determine it:

• Simply subtract any term from the subsequent term. In our example, subtract 5 from 7 to find \( d = 2 \).

This difference of 2 means that every term is exactly 2 units apart from the preceding one. Understanding the common difference tells us how the sequence grows or shrinks depending on whether the value is positive or negative. Here, it's steadily increasing. Recognizing this pattern facilitates the use of relevant arithmetic sequence formulas and reinforces the predictable nature of such sequences.

Sequence Terms

Sequence terms are the individual numbers that make up an arithmetic sequence. They follow a specific pattern dictated by the common difference. Every term can be found using the general formula for an arithmetic sequence:\[a_n = a + (n-1)d\]Each term \( a_n \) is determined by starting with the first term \( a \) and adding the common difference \( d \) a certain number of times. For the 50th term, for example, you could apply:\[a_{50} = 5 + (50 - 1) \times 2 = 103\]By using this formula, you can predict and verify any term's value within the sequence. This method ensures an efficient way to manage and understand the arithmetic progression beyond just listing them out.