Problem 53 - Precalculus Mathematics for Calculus

Question

A partial sum of an arithmetic sequence is given. Find the sum. $$\sum_{k=0}^{10}(3+0.25 k)$$

Step-by-Step Solution

Answer

The sum is 46.75.

Step 1: Identify the Arithmetic Sequence

The given expression represents the sum of an arithmetic sequence where each term can be written as \( a_k = 3 + 0.25k \). Here, \(a_0\) is the first term and \(d = 0.25\) is the common difference.

Step 2: Determine the Number of Terms

Since the summation index \(k\) starts at 0 and goes to 10, there are 11 terms in the sequence. Thus, \(n = 11\).

Step 3: Calculate the First Term (\(a_1\)) and Last Term (\(a_n\))

The first term, \(a_1\), is calculated when \(k=0\): \[ a_1 = 3 + 0.25(0) = 3 \].The last term, \(a_n\), is calculated when \(k=10\): \[ a_n = 3 + 0.25(10) = 5.5 \].

Step 4: Use the Arithmetic Sum Formula

The sum \(S_n\) of an arithmetic sequence is given by the formula:\[ S_n = \frac{n}{2} (a_1 + a_n) \]. Substituting the known values, \[ S_{11} = \frac{11}{2} (3 + 5.5) \].

Step 5: Calculate the Sum

Calculate \(S_{11}\) by substituting the values into the formula.First, compute the sum of the first and last terms: \(3 + 5.5 = 8.5\).Then, multiply by \(\frac{11}{2}\): \[ S_{11} = \frac{11}{2} \times 8.5 = \frac{11 \times 8.5}{2} = \frac{93.5}{2} = 46.75 \].

Key Concepts

Partial Sum

When you're working with arithmetic sequences, you'll often encounter the concept of a partial sum. A partial sum is the sum of a specific number of terms from the sequence. In the problem at hand, we are calculating the sum of the first 11 terms of the sequence given by \( a_k = 3 + 0.25k \). Remember, a partial sum isn't the total sum of an infinite sequence—it only includes a certain number of its initial terms.
This is crucial for understanding how sequences develop over a finite range, and it helps in identifying patterns or deriving formulas for more complex calculations. When calculating a partial sum, make sure you have identified all relevant terms, including both the first and last terms you wish to sum.

Common Difference

In arithmetic sequences, the "common difference" is a key component that defines the sequence. It is the constant amount added to each term to produce the next one. In the sequence \( a_k = 3 + 0.25k \), the common difference \( d \) is \( 0.25 \).
This means that each term is 0.25 greater than the one before it.

• Start with the first term: \( a_0 = 3 \).
• Add the common difference repeatedly: \( 3.25, 3.5, 3.75, \) and so on.

Recognizing the common difference helps predict future terms and forms the foundation for calculating the total sum of several terms in a sequence.

Arithmetic Sum Formula

The arithmetic sum formula is essential when you want to quickly calculate the sum of terms in an arithmetic sequence. For an arithmetic sequence with \( n \) terms, the sum \( S_n \) is given by:\[S_n = \frac{n}{2} (a_1 + a_n)\]
This formula simplifies the process, enabling you to find the total sum without adding each term individually. In our scenario, we used \( n = 11 \), \( a_1 = 3 \), and \( a_n = 5.5 \) to find the sum \( S_{11} = 46.75 \).

• Identify the number of terms: Here, \( n = 11 \).
• Calculate the first term, \( a_1 \), and the last term, \( a_n \).
• Apply these values to the formula for an efficient solution.

Mastering this formula can save you time and significantly reduce the possibility of errors in arithmetic sum calculations.

Sequence Terms

The term "sequence terms" refers to the individual elements within a sequence. These terms are arranged in a specific order following a particular rule or formula. For our arithmetic sequence, each term \( a_k \) is determined by the formula \( 3 + 0.25k \).
Each new term comes from plugging consecutive integer values for \( k \) starting from 0, leading to terms like 3, 3.25, 3.5, etc.

• First term: \( a_0 = 3 \) (when \( k = 0 \)).
• Apply the rule to find subsequent terms.

Understanding sequence terms is vital for grasping the broader pattern of the sequence, making it easier to solve related problems and explore more advanced concepts in sequences.