Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=8, d=3$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 215.

Step 1: Identify the Formula

The sum of the first n terms of an arithmetic sequence can be calculated using the formula: \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( d \) is the common difference.

Step 2: Substitute the Values

Substitute the given values into the formula: \( n = 10 \), \( a_1 = 8 \), and \( d = 3 \) into the formula: \[ S_{10} = \frac{10}{2} (2 \cdot 8 + (10-1) \cdot 3) \].

Step 3: Simplify the Equation

First, calculate inside the parentheses: \( 2 \cdot 8 = 16 \) and \( (10-1) \cdot 3 = 27 \). So, we have \( 16 + 27 = 43 \).

Step 4: Calculate the Sum

Now substitute back into the formula: \( S_{10} = 5 \times 43 = 215 \).

Key Concepts

Sum of Arithmetic Sequence

When dealing with arithmetic sequences, it's often necessary to calculate the sum of a certain number of terms. This is known as the "Sum of Arithmetic Sequence". The formula to find this sum is: \[ S_n = \frac{n}{2} \times (2a_1 + (n-1) \cdot d) \] where

• \( S_n \) represents the sum of the first \( n \) terms,
• \( a_1 \) stands for the first term of the sequence,
• \( d \) is the common difference, which is the amount that each term increases by,
• and \( n \) is the total number of terms you wish to sum.

To apply this formula, substitute the known values for \( a_1 \), \( d \), and \( n \), and then perform the arithmetic operations according to the order of operations (parenthesis, multiplication, and addition). This will yield the total sum of the arithmetic sequence for the specified number of terms.

First Term of Sequence

The "First Term of Sequence", often denoted as \( a_1 \), is the starting point or the initial term of an arithmetic sequence. For any arithmetic sequence, the first term sets the base from which each subsequent term is calculated by adding the common difference. In the example problem, the first term of the sequence is given as 8. This means that the sequence starts at 8, and each following term is determined by adding the common difference to this first term. Understanding the first term is crucial because it helps define the behavior and values of the rest of the sequence.

Common Difference

The "Common Difference" in an arithmetic sequence is a key concept, as it determines the interval between each term. Denoted as \( d \), this value is what you add to each term to get the next one.For instance, in the provided example, the common difference is 3. This implies that starting from the first term (8), each subsequent term will be obtained by adding 3 to the current term.

• If \( d \) is positive, the sequence will increase, adding more to each term as you move along the sequence.
• If it's negative, the sequence will decrease, subtracting from each term.

Calculating terms continuously using this common difference helps form the entire sequence and is vital in finding the sum of a sequence because it impacts the total change accumulated over all terms.

Number of Terms

The "Number of Terms" in an arithmetic sequence is another foundational component, represented by \( n \). This tells you how many terms from the starting point to include when calculating the sum or solving related problems. In this context, \( n = 10 \), meaning the sequence's sum involves the first ten terms. It sets the bounds on calculations for both individual term identification and summing purposes.

• The larger the \( n \), the greater the sequence will be, and thus the sum.
• Ensuring the correct value of \( n \) is used is critical for accurate results.

Identifying and understanding these terms helps manage expectations about the sequence's size and the resultant calculations.