Problem 9 - Algebra and Trigonometry

Question

\(9-12\) . Find the \(n\) th term of the arithmetic sequence with given first term and common difference \(d\) What is the 10 the term? $$ a=3, d=5 $$

Step-by-Step Solution

Answer

The 10th term is 48.

Step 1: Identify the formula for the nth term

To find the nth term of an arithmetic sequence, use the formula: \( a_n = a + (n-1) \cdot d \), where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

Step 2: Plug in the given values

Here, the first term \( a = 3 \) and the common difference \( d = 5 \). You need to find the 10th term, so \( n = 10 \). Substitute these values into the formula: \( a_{10} = 3 + (10-1) \cdot 5 \).

Step 3: Perform the calculation inside the parentheses

First, calculate \( 10 - 1 \) which equals 9. So the modified equation is \( a_{10} = 3 + 9 \cdot 5 \).

Step 4: Multiply the common difference by \( n-1 \)

Now, calculate \( 9 \cdot 5 \) which equals 45. Substitute this back into the equation: \( a_{10} = 3 + 45 \).

Step 5: Add the results to find the nth term

Add 3 and 45 to find the 10th term. Thus \( a_{10} = 48 \).

Key Concepts

nth term formula

In any arithmetic sequence, the nth term formula is crucial for determining any term within the sequence. The general formula for the nth term is given by:

• \( a_n = a + (n-1) \cdot d \)

Here, \(a\) represents the first term of the sequence, \(d\) is the common difference, and \(n\) is the position of the term you wish to find. This formula helps to calculate the term by starting from the initial term \(a\) and adding the common difference \(d\) a specific number of times, precisely \(n-1\) times.
This means, for example, if you have a sequence starting with 3, and a common difference of 5, you can find the 10th term by using this formula: plug in \(a = 3\), \(d = 5\), and \(n = 10\). This allows you to systematically find any term you need without having to list all terms up to that point.

common difference

An arithmetic sequence is characterized by its common difference, a constant value that separates each term from the next. This difference is represented by \(d\) in the formula, \(a_n = a + (n-1) \cdot d\).
The common difference tells us how much we add (or subtract if it's negative) to any term to get to the next term. Understanding this concept gives insights into how the sequence grows or shrinks as you progress through its terms.
For example, in the sequence defined by \(a = 3\) and \(d = 5\), you start with the number 3 and add 5 to each subsequent term. So the sequence begins as 3, 8, 13, 18, and so on. The common difference \(d = 5\) controls this consistent increment.

arithmetic progression

An arithmetic progression, or arithmetic sequence, involves a series of numbers arranged in a specific pattern where each term after the first is created by adding a constant, known as the common difference.
Arithmetic progressions are linear in nature, meaning they increase (or decrease) steadily as you progress through the terms. This property makes them predictable and easy to analyze using the nth term formula.

• Simple structure: a fixed increase or decrease each time.
• Allows for easy term calculations using formulas.

In our example with \(a = 3\) and \(d = 5\), the sequence 3, 8, 13, etc., represents a clear arithmetic progression where each number is formed by adding 5 to its predecessor.

sequence term calculation

Calculating a specific term within an arithmetic sequence utilizes the nth term formula effectively. Once you understand the concepts of the common difference and the arithmetic progression, finding a particular term becomes straightforward.
Start by identifying the first term and the common difference, then decide which term you need, referred to as \(n\). By applying the formula \( a_n = a + (n-1) \cdot d \), you can substitute these values to find the desired term.

• Identify \(a\), \(d\), and \(n\).
• Substitute into the formula.
• Calculate \(n-1\) and multiply by \(d\).
• Add the result to \(a\).

For instance, in our narrative, the 10th term calculation starts with identifying \(a = 3\), \(d = 5\), and \(n = 10\), leading to \( a_{10} = 3 + (10-1) \cdot 5 = 3 + 45 = 48\). This step-by-step approach ensures you can determine any term with ease.