Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-22: Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$ a_{1}=-9, \quad a_{10}=15, \quad n=10 $$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 30.

Step 1: Understand the Arithmetic Sequence

Given the first term of an arithmetic sequence, \(a_1 = -9\), and the 10th term, \(a_{10} = 15\), with \(n = 10\), we need to find the sum of the first 10 terms of this sequence.

Step 2: Use the Formula for the nth Term

The nth term of an arithmetic sequence can be calculated using the formula \(a_n = a_1 + (n-1) \cdot d\), where \(d\) is the common difference. We have \(a_1 = -9\) and \(a_{10} = 15\), so \[ 15 = -9 + (10-1) \cdot d \]Simplifying gives: \[ 15 = -9 + 9d \]

Step 3: Solve for the Common Difference

Now solve for \(d\) by first adding 9 to both sides: \[ 15 + 9 = 9d \]\[ 24 = 9d \]Divide both sides by 9 to find \(d\): \[ d = \frac{24}{9} = \frac{8}{3} \]

Step 4: Find the Sum of the First n Terms

The sum of the first \(n\) terms of an arithmetic sequence is given by the formula \[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]For this problem, \(n = 10\), \(a_1 = -9\), and \(a_{10} = 15\).Substitute these values into the formula: \[ S_{10} = \frac{10}{2} \cdot (-9 + 15) \]Simplify: \[ S_{10} = 5 \cdot (6) \]\[ S_{10} = 30 \]

Key Concepts

Common Difference

The concept of the 'Common Difference' is key to understanding arithmetic sequences. Imagine an arithmetic sequence as a chain of numbers where each number is separated by the same step, known as the common difference. This difference is represented by the symbol \(d\). It is the amount you add to each term to get to the next one.
In our exercise, we have an arithmetic sequence starting at \(a_1 = -9\) and going up to \(a_{10} = 15\). We use the nth term formula, \(a_n = a_1 + (n-1) \cdot d\), to identify the common difference. By plugging in the values, we deduce that \(d = \frac{8}{3}\).

• This means each subsequent term increases by \(\frac{8}{3}\).
• Understanding the common difference helps in quickly predicting further terms or calculating totals in the sequence.

Sum of an Arithmetic Sequence

An arithmetic sequence involves adding a series of terms where each moves forward by a consistent step. The 'Sum of an Arithmetic Sequence' helps determine the total when you add up these terms. Knowing the common difference is just a starting point; we seek the entire sum over a range of the sequence.
In our problem, you want the sum of the first 10 terms. The formula for this is \( S_n = \frac{n}{2} \cdot (a_1 + a_n) \). It uses the whole sequence by connecting the first and the nth term. For example, here we have \(n = 10\), \(a_1 = -9\), and \(a_{10} = 15\). Substituting the numbers gives a sum of 30.

• This formula helps collapse potentially complex arithmetic into a simple operation.
• It finds utility in numerous practical contexts, like calculating distances, costs, or even total growth over time.

nth Term Formula

The 'nth Term Formula' is fundamental for pinpointing any term in an arithmetic sequence without having to list all previous terms. The formula \( a_n = a_1 + (n-1) \cdot d \) serves as your shortcut to jump directly to the desired term.
In practical terms, if you know where your sequence starts \((a_1)\) and the step \((d)\), you can find terms quickly. In our exercise, to locate the 10th term, we fit the numbers into the formula. Knowing \(a_1 = -9\) and \(d = \frac{8}{3}\), finding \(a_{10}\) involves some simple arithmetic, confirming \(a_{10} = 15\).

• This formula aids in solving sequences quicker and is the backbone of many calculations involving arithmetic progression.
• It's particularly useful in forecasting, budgeting, or any area where sequences figure prominently.