Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated \(n\) th partial sum of the arithmetic sequence. $$4.2,3.7,3.2,2.7, . . . ; n=12$$

Step-by-Step Solution

Answer

The 12th partial sum of the arithmetic sequence is 17.4

Step 1: Identifying the sequence parameters

First, recognize that it's an arithmetic sequence where each term decreases by 0.5. The first term, labelled \(a_1\), is 4.2. The common difference, labelled \(d\), is -0.5. The number of terms, labelled \(n\), is 12.

Step 2: Applying the formula for the n-th partial sum of an arithmetic sequence

To find the partial sum of the first 12 terms of the sequence, apply the formula: \(S_n = n/2 [2a_1 + (n-1)*d]\). Substitute the specified values into the equation to obtain: \(S_{12} = 12/2 * [2*4.2 + (12-1)*-0.5]\).

Step 3: Simplifying the expression

Carrying out the calculations, we get: \(S_{12} = 6 * [8.4 - 5.5] = 6 * 2.9\). Multiplying these gives \(S_{12} = 17.4\) which is the 12th partial sum of the sequence.

Key Concepts

Partial Sum

Calculating the partial sum of an arithmetic sequence involves adding up the first few terms of the sequence up to the given position. This is a helpful concept when you want to find the cumulative effect of a sequence without having to add each term manually.

For arithmetic sequences, the formula to calculate the partial sum, denoted as \(S_n\), is given by:

• \(S_n = \frac{n}{2} [2a_1 + (n-1) \cdot d]\)

Here, \(n\) is the number of terms, \(a_1\) is the first term of the sequence, and \(d\) is the common difference between terms.

This formula efficiently sums up the sequence by leveraging the properties of arithmetic sequences, making it much faster than adding each term one by one. By applying this formula to the problem, we calculated the sum of the first 12 terms as 17.4.

Common Difference

The common difference is a key feature of arithmetic sequences. It represents the constant change between consecutive terms. This value can be positive, negative, or even zero, affecting whether the sequence increases, decreases, or remains constant.

The common difference is denoted by \(d\) and can be found by subtracting any term from the term that follows it. For example, in our sequence:

• \(d = 3.7 - 4.2 = -0.5\)

Understanding the common difference allows you to predict any term within a sequence easily. When dealing with such sequences, always ensure to consistently apply \(d\) to identify patterns or solve for unknown terms.

In our exercise, since \(d\) is -0.5, it indicates that each subsequent term decreases by 0.5 from the previous term.

Sequence Formula

The sequence formula is an essential tool when dealing with arithmetic sequences. It helps in locating any specific term in the sequence quickly. The general form of the sequence formula for an arithmetic sequence is:

• \(a_n = a_1 + (n-1) \cdot d\)

Here, \(a_n\) is the \(n\)-th term you wish to find, \(a_1\) is the very first term, \(n\) is the term's position in the sequence, and \(d\) is the common difference.

This formula is useful because it establishes a straightforward process for finding any term without listing every preceding one. For our given sequence, using this formula, you can predict terms further down the line.

Such formulas are foundational tools in understanding arithmetic sequences, helping to simplify and solve complex problems with ease.