Problem 40 - Algebra and Trigonometry

Question

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=3, d=2, n=12$$

Step-by-Step Solution

Answer

The partial sum of the first 12 terms is 168.

Step 1: Identify the formula for the nth term

The nth term in an arithmetic sequence can be calculated using the formula \(a_n = a + (n-1)d\). This formula allows you to find any term in the sequence given the first term \(a\), the common difference \(d\), and the term number \(n\).

Step 2: Calculate the 12th term

Using the formula for the nth term, \(a_n = a + (n-1)d\), substitute \(a=3\), \(d=2\), and \(n=12\) to find \(a_{12}\):\[a_{12} = 3 + (12-1) \cdot 2\]\[a_{12} = 3 + 22 = 25\].Thus, the 12th term is 25.

Step 3: Use the formula for the sum of the first n terms

The sum of the first n terms \(S_n\) of an arithmetic sequence can be calculated as \(S_n = \frac{n}{2} \cdot (a + a_n)\). This formula uses the number of terms \(n\), the first term \(a\), and the nth term \(a_n\).

Step 4: Substitute the known values to find the partial sum

Substitute \(n=12\), \(a=3\), and \(a_{12}=25\) into the sum formula:\[S_{12} = \frac{12}{2} \cdot (3 + 25)\]\[S_{12} = 6 \cdot 28\]\[S_{12} = 168\].The sum of the first 12 terms is 168.

Key Concepts

Partial Sum

In an arithmetic sequence, a partial sum is the sum of a certain number of terms from the beginning of the sequence. It helps you see the accumulation of values as you move through the sequence. The formula for the partial sum, usually denoted as \(S_n\), is:

• \( S_n = \frac{n}{2} \cdot (a + a_n) \)

This formula involves the total number of terms you want to add, \(n\), the first term \(a\), and the nth term \(a_n\). By calculating the partial sum, you can determine how much these terms add up to within a particular stretch of the sequence. In our example, the partial sum of the first 12 terms is calculated to be 168.

Nth Term

The nth term of an arithmetic sequence is used to find any term within the sequence. It is a specific element with a position number \(n\). This term can be calculated using the formula:

• \( a_n = a + (n-1) \cdot d \)

Here, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the position within the sequence. By substituting these values, you can find out what number sits at any given place. In our example for \(n = 12\), the 12th term \(a_{12}\) is 25.

Common Difference

The common difference is a key feature of an arithmetic sequence. It is the constant difference between consecutive terms in the sequence, and it defines how the sequence progresses. In a formulaic sense, it's expressed as:

• \( d = a_{n} - a_{n-1} \)

This means each term is a fixed amount (\(d\)) more than the previous one. In our given sequence, the common difference \(d\) is 2, showing that each term increases by 2 from the one before.

Sum of Terms

The sum of terms refers to the result you get when you add a specified number of terms within an arithmetic sequence. This sum offers a way to understand how values accumulate in the sequence. To find the sum of the first \(n\) terms (

• \( S_n = \frac{n}{2} \cdot (2a + (n-1)d) \)

Alternatively, you can use \( S_n = \frac{n}{2} \cdot (a + a_n) \). This involves knowing the first term \(a\), the number of terms \(n\), and either the nth term or the common difference. For our sequence, the sum of the first 12 terms was calculated as 168, showing how the values increase collectively.