Problem 35 - Algebra 2

Question

Evaluate each series to the given term. \(2+4+6+8+\ldots ; 10\) th term

Step-by-Step Solution

Answer

The sum of the series up to the 10th term is 110.

Step 1: Identify the common difference (d) and the first term (a)

We have an arithmetic sequence starting from 2 and the subsequent terms increase by 2. Therefore, the common difference (d) is 2, and the first term (a) is 2.

Step 2: Find the 10th term (l)

The formula to find the nth term in an arithmetic sequence is \(a + (n - 1) * d\). Let's substitute n = 10, a = 2, and d = 2 into the equation. The result is \(2 + (10 - 1)*2 = 20\). Therefore, the 10th term (l) in the series is 20.

Step 3: Evaluate the sum of the series up to the 10th term

We have identified the number of terms (n) as 10, the first term (a) as 2 and the last term (l) as 20. Using the formula \(S = n/2 * (a + l)\), we get \(S = 10/2 * (2 + 20) = 110\).

Key Concepts

Understanding the Common Difference

In arithmetic sequences, one of the key characteristics is the common difference. It is denoted as "d." The common difference is a constant that you add to each term to get to the next term in the sequence. This feature makes arithmetic sequences predictable and easy to work with.

• In our example sequence, the common difference is 2. We found this by subtracting the first term from the second term: 4 - 2 = 2.
• Every time we move from one term to the next, we increase by this number. For instance, starting from 2, the sequence becomes 2, 4, 6, 8, and so on.

Knowing the common difference is crucial for calculating other elements of the sequence, such as individual terms and the total sum.

Using the nth Term Formula

The nth term formula in an arithmetic sequence helps you find the value of any term, given its position in the sequence. This is especially useful when dealing with large sequences, where counting by hand is inefficient.
The formula is:\[ a_n = a + (n - 1) \cdot d \]

• Here, \( a \) is the first term, \( n \) is the term position you want to find, and \( d \) is the common difference.
• In our example, we wanted to find the 10th term. Substituting the values: \( a = 2 \), \( n = 10 \), and \( d = 2 \), the calculation is \( 2 + (10 - 1) \times 2 = 20 \). Therefore, the 10th term is 20.

Understanding this formula is essential for finding terms efficiently in an arithmetic sequence.

Series Evaluation in Arithmetic Sequences

Series evaluation involves calculating the sum of a sequence of numbers that follow a pattern. In an arithmetic sequence, evaluating the series means finding the sum of its terms up to a specific point, such as the 10th term.

• Initially, identify the necessary components like the number of terms \( n \), the first term \( a \), the common difference \( d \), and the last term \( l \) you are adding up to.
• In our exercise, the identified components were: \( a = 2 \), \( n = 10 \), and \( l = 20 \).

The process of a series evaluation makes use of these components for efficient calculation of the total sum.

Calculating the Sum of an Arithmetic Sequence

The formula for calculating the sum of an arithmetic sequence is a handy tool to quickly add up a series. The formula is:\[ S_n = \frac{n}{2} \cdot (a + l) \]Here:

• \( S_n \) is the sum of the first \( n \) terms.
• \( n \) is the number of terms.
• \( a \) is the first term, and \( l \) is the last term you are summing up to.

In the exercise, the sum of the first 10 terms of the sequence was calculated by substituting \( n = 10 \), \( a = 2 \), and \( l = 20 \) into the formula. Performing the calculation gives us:\[ S_{10} = \frac{10}{2} \times (2 + 20) = 110 \]Thus, the sum of the sequence up to the 10th term is 110. This method makes it simple to find the total of long sequences without adding each element individually.