Question

Among the following, which term belongs to the arithmetic progression \(-5,2,9, \ldots ?\) (1) 342 (2) 343 (3) 344 (4) 345

Short answer

Solution: Analyzing the arithmetic progression, we find that two terms, 342 and 344, belong to the given arithmetic progression.

Step-by-step solution

Identify the common difference

In an arithmetic progression, the common difference is the difference between two consecutive terms. So, by subtracting the preceding term from the succeeding term, we can get the common difference: \(d = 2 - (-5) = 7\)

Write the general form for the given arithmetic progression

The general form for arithmetic progression is: \(a_n = a_1 + (n - 1) \times d\) In this case, \(a_1 = -5\) and the common difference \(d = 7\). So the formula becomes: \(a_n = -5 + (n - 1) \times 7\)

Check for each term if it can be represented by the formula

To check if any of the given options is a term of the arithmetic progression, we simply need to substitute \(a_n\) with the given term and see whether the equation has a natural number solution for \(n\) or not. 1) For \(a_n = 342\), \(n = \frac{342 + 5}{7} + 1 = 50\). So, first term is a part of AP. 2) For \(a_n = 343\), \(n = \frac{343 + 5}{7} + 1 = \frac{348}{7} + 1 = 50.5714286 \ldots\). Not a natural number, thus second term is not part of AP. 3) For \(a_n = 344\), \(n = \frac{344 + 5}{7} + 1 = \frac{349}{7} + 1 = 51\). So, third term is a part of AP. 4) For \(a_n = 345\), \(n = \frac{345 + 5}{7} + 1 = \frac{350}{7} + 1 = 51.5714286 \ldots\). Not a natural number, thus fourth term is not part of AP. Hence, among the given options, two terms, 342 and 344 belong to the given arithmetic progression.

Key concepts

Common Difference

The term 'common difference' is a critical component in understanding arithmetic progressions. It refers to the uniform interval that separates consecutive terms in an arithmetic sequence, and it is what defines the sequence's linearity. Essentially, it is the constant value you add (or subtract if the value is negative) to get from one term to the next.

To identify the common difference in the exercise, one simply calculates the difference between two adjacent terms. For instance, given the sequence \( -5, 2, 9, \ldots \), the difference between the second term, 2, and the first term, -5, presents us with the common difference: \( d = 2 - (-5) = 7 \). Understanding this concept is pivotal for students to grasp as it allows them to determine whether a given number is part of a specific arithmetic sequence or not, as well as construct subsequent terms.

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a distinct property: each term after the first is derived by adding a constant number, known as the common difference, to the previous term. This series of numbers is a foundational concept in algebra and is characterized by its predictable pattern.

In the provided exercise, the series \( -5, 2, 9, \ldots \) is an example of an arithmetic sequence where the common difference is 7. To confirm if a number belongs to this sequence, we deploy the general arithmetic sequence formula: \(a_n = a_1 + (n - 1) \times d\), where \(a_n\) represents the nth term in the sequence, \(a_1\) is the first term, and \(d\) is the common difference. By substituting in different values of 'n', we identify which natural numbers will yield a term within the sequence.

Natural Number

Natural numbers are the building blocks of arithmetic; they consist of the counting numbers beginning from 1 and extending indefinitely (1, 2, 3, ...). Importantly, they do not include negative numbers or fractions—this characteristic is pertinent when determining if an element belongs to an arithmetic sequence.

When we solve for 'n' in our sequence formula—like in the exercise's step-by-step solution—if the result is a natural number, it means that we have successfully found the position of the term in the sequence. If the number is not a natural number (for example, it's a fraction or a negative number), the term does not belong to the sequence. Therefore, from the multiple-choice options given, we can assert that only certain values correspond to a natural number position in the sequence, thus affirming their membership in the arithmetic progression.