Problem 40 - Precalculus: Mathematical for Calculus

Question

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 \( S_{12} \) is 168.

Step 1: Identify the General Formula

For an arithmetic sequence, the nth term is found using the formula: \( a_n = a + (n-1)d \). We will use this formula to find the 12th term.

Step 2: Calculate the Last Term \(a_{12}\)

Substitute the values into the formula to find the 12th term: \( a_{12} = 3 + (12-1) \cdot 2 = 3 + 22 = 25 \). Thus, the 12th term is 25.

Step 3: Use the Formula for Partial Sum

The sum of the first n terms of an arithmetic sequence is given by: \( S_n = \frac{n}{2} (a + a_n) \). This formula will allow us to calculate the sum of the first 12 terms.

Step 4: Substitute Values into the Sum Formula

Insert the known values into the sum formula: \[ S_{12} = \frac{12}{2} (3 + 25) = 6 \cdot 28 \].

Step 5: Calculate the Partial Sum \(S_{12}\)

Multiply the values inside the parentheses to get: \( 6 \cdot 28 = 168 \). Therefore, the partial sum \( S_{12} \) is 168.

Key Concepts

Partial Sum

In an arithmetic sequence, a partial sum refers to the sum of a specific number of terms from the beginning of the sequence. This is a useful concept when you want to add up the first few elements rather than the entire sequence. The formula used is:

• \( S_n = \frac{n}{2} (a + a_n) \)

This formula helps us calculate the sum of the first \( n \) terms, where \( a \) is the first term and \( a_n \) is the nth term. For example, to find the partial sum of the first 12 terms in the sequence from our exercise, we first identify the 12th term and then apply the formula. This step-by-step process ensures accuracy while using the partial sum formula. Calculating partial sums is particularly beneficial in problems involving financial applications, like calculating interest over a period of time.

Nth Term Formula

The nth term formula is fundamental in defining an arithmetic sequence. It helps us determine any term in the sequence without listing all prior terms. The formula is:

• \( a_n = a + (n-1)d \)

Here, \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number. For our exercise, to find the 12th term, we substitute the given values (\( a = 3 \), \( d = 2 \), and \( n = 12 \)) into the formula. This provides us with \( a_{12} = 3 + (12-1) \cdot 2 = 25 \). Knowing the nth term allows us not only to identify any specific term but also plays an integral role when calculating the partial sum in the sequence.

Common Difference

The common difference is a key characteristic of arithmetic sequences. It represents the constant amount added to each term to get to the next term. Given as \( d \), the common difference can be positive, negative, or zero. In our exercise, the common difference is \( d = 2 \), indicating that each term increases by 2 from the previous term. Understanding this helps clarify why an arithmetic sequence progresses linearly and aids in using both the nth term formula and partial sum formula. It simplifies calculations and makes it easier to predict subsequent terms or the sum of a set of terms in the sequence.