Problem 33 - A Graphical Approach to College Algebra

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: Understand the Formula

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

Step 2: Identify Known Values

We know \( a_1 = 8 \), \( d = 3 \), and \( n = 10 \). These values will be plugged into the sum formula.

Step 3: Apply Formula

Substitute the known values into the formula: \[ S_{10} = \frac{10}{2} \times (2 \times 8 + (10-1)\times 3). \]

Step 4: Simplify the Inner Expression

Calculate inside the brackets: \[ 2 \times 8 = 16, \quad (10-1) \times 3 = 27. \]So, replace the bracketed part with:\[ 2 \times 8 + (10-1)\times 3 = 16 + 27 = 43. \]

Step 5: Calculate the Sum

Complete the calculation:\[ S_{10} = \frac{10}{2} \times 43 = 5 \times 43 = 215. \]

Key Concepts

Sum of Sequence

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. To find the sum of the first few terms in an arithmetic sequence, you can use the sum formula: \[ S_n = \frac{n}{2} \times (2a_1 + (n-1)d) \]Here, \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the initial term, \( d \) is the common difference (the amount added to each term to get the next one), and \( n \) is the number of terms you want to add up.
Let's break down what the formula means. The \( \frac{n}{2} \) part of the formula is there because it's the average of the terms when you add the first term and the last term. The expression \((2a_1 + (n-1)d)\) calculates the sum of the first and last term in the range of terms you're considering. Multiplying by \( \frac{n}{2} \) then gives the total sum of the sequence.

Common Difference

The common difference \( d \) in an arithmetic sequence is the amount by which we increase (or decrease, if negative) to get from one term to the next.
If you start with an initial term \( a_1 \) and have a common difference \( d \), each term after the first can be found by adding \( d \) to the previous term.
For example, with \( a_1 = 8 \) and \( d = 3 \), the sequence would go: 8, 11, 14, 17, 20, and so on. Every term increases by 3 compared to the one before it.

• If \( d \) is positive, the sequence will increase.
• If \( d \) is negative, the sequence will decrease.

Understanding the common difference is essential for forming the sequence and finding specific terms.

Terms of Sequence

In arithmetic sequences, each element is called a term. The terms are systematically arranged based on a starting point and a common difference.
The formula used to find any term in an arithmetic sequence is \( a_n = a_1 + (n-1)d \). Here, \( a_n \) is the \( n \)th term you want to find. Knowing this formula allows you to find any term in the sequence without listing all the terms before it.
Let’s consider our sequence where \( a_1 = 8 \), \( d = 3 \), and \( n = 10 \). To find the 10th term, substitute the known values into the formula:\[ a_{10} = 8 + (10-1) \times 3 = 8 + 27 = 35 \]Thus, the 10th term is 35. This approach is particularly useful when dealing with long sequences, where listing each individual term would be time-consuming.