Question

Find the ninth term of the AP $$x, \quad x+3 y, \quad x+6 y, \ldots$$

Short answer

The ninth term of the given AP is 'x + 24y'.

Step-by-step solution

Identify the first term

The first term of an arithmetic progression (AP) is the starting point of the sequence. In the given AP, the first term is denoted by 'x'. Therefore, the first term (a_1) is 'x'.

Determine the common difference

The common difference (d) of an AP is the difference between any two consecutive terms. By subtracting the first term from the second term, we have: (x + 3y) - x = 3y. Therefore, the common difference 'd' is '3y'.

Use the general term formula for an AP

The nth term (a_n) of an AP is obtained using the formula a_n = a_1 + (n - 1)d. To find the 9th term, we substitute 'n' with 9 in the formula.

Calculate the ninth term

Substituting the given values into the nth term formula, we get a_9 = x + (9 - 1)(3y) = x + 8(3y) = x + 24y. Thus, the ninth term of the given AP is 'x + 24y'.

Key concepts

First Term of an AP

Understanding the first term of an arithmetic progression (AP) is crucial because it serves as the foundation from which the entire sequence is built. In simple terms, the first term, often represented by the symbol 'a', is the starting value of the sequence. For example, if we're looking at a list of numbers that starts with 5, like 5, 7, 9..., then 5 is our first term, denoted by 'a_1'. This sets the stage for determining the remaining elements of the AP by adding the common difference to the first term repeatedly.

When interpreting the first term in the context of an exercise, be precise about the given elements. If the sequence starts with a variable, such as 'x' in the sequence given by the exercise \(x, x+3y, x+6y, \ldots\), 'x' is indeed the first term of that AP. Recognizing this correctly paves the way for further calculations within the sequence.

Common Difference in an AP

The common difference of an arithmetic progression is perhaps one of the most pivotal concepts to grasp. It is the uniform amount by which consecutive terms increase (or decrease) and is denoted by 'd'. To find the common difference, simply subtract the first term from the second term. For the sake of illustration, if we have an AP where the subsequent term after 5 is 7, our common difference is 7 - 5, which equals 2.

In the given exercise, to find the common difference 'd', we take the second term \(x + 3y\) and subtract the first term 'x', resulting in \(3y\). This value is the step we consistently add to each term to reach the next one in the progression. It's this uniform stepping that defines the nature of an arithmetic sequence and governs its behavior throughout.

General Term Formula of an AP

The general term formula of an AP is a mathematical representation that provides us with a method to find any term within an arithmetic progression, without having to list out all the preceding terms. This formula is expressed as \(a_n = a_1 + (n - 1)d\), where \(a_n\) is the nth term we want to find, \(a_1\) is the first term, 'n' is the term's position in the sequence, and 'd' is the common difference.

With this powerful formula, solving for the ninth term—or any term—becomes a straightforward task of substitution. In our exercise, to find the ninth term \(a_9\), we substitute 'n' with 9 and use the identified first term 'x' and common difference \(3y\) from earlier steps. Calculating \(a_9 = x + (9 - 1)(3y)\), simplifies to \(x + 24y\), delivering the ninth term efficiently.

Sequence and Series

Sequences and series are foundational concepts in arithmetic and beyond. A sequence is an ordered list of numbers following a particular rule, such as an arithmetic progression where each term is created by adding a constant value, the common difference. A series, on the other hand, is the sum of the terms of a sequence. If we were to add all the terms in our AP sequence, we'd be dealing with an arithmetic series.

In practical terms, understanding sequences helps us to predict and find specific values within a list of numbers, while comprehension of series can be employed to calculate totals over a range of terms. Both sequences and series play pivotal roles in various fields including finance, computer science, and physics, enhancing the importance of understanding these fundamental concepts.