Question

Which term of the arithmetic progression \(21,42,63,84, \ldots\) is \(420 ?\) (1) 19 (2) 20 (3) 21 (4) 22

Short answer

Answer: 20

Step-by-step solution

Identify the first term and common difference

In the arithmetic progression \(21, 42, 63, 84, \ldots\), the first term is \(a=21\). To find the common difference, subtract the first term from the second term, so \(d=42-21=21\).

Use the arithmetic progression formula

Use the arithmetic progression formula to find the term number: \(T_n = a + (n - 1)d\). We are given \(T_n=420\), so we need to solve for n. Substitute the given values into the equation: \(420 = 21 + (n - 1)21\).

Solve for n

Simplify and solve for n: \(420 = 21 + 21(n - 1)\) \(420 = 21 + 21n - 21\) \(420 = 21n\) \(n = \frac{420}{21}\) \(n = 20\) The term number is 20. The solution to the problem is option (2) 20.

Key concepts

Understanding the First Term in Arithmetic Progression

In any arithmetic progression (AP), the sequence begins with what we call the "first term." This term is crucial because it forms the starting point of your sequence. Think of it as the very first step you take on a staircase. In our example, the sequence is 21, 42, 63, 84, etc., and the first term, represented by \( a \), is 21.
When solving problems related to arithmetic progressions, identifying the first term is critical. This first term often gets directly plugged into formulas and equations. For any AP, you can quickly find your starting point by simply observing the very first number in the sequence. Remember, without this initial term, it's impossible to accurately describe or calculate the rest of the sequence.

The Role of Common Difference in Arithmetic Progression

The common difference is what makes your arithmetic progression move forward. It's the amount by which each term increases over the previous one, keeping the sequence consistent. For the sequence 21, 42, 63, 84, etc., your common difference, denoted by \( d \), is 21.

• To find the common difference, you subtract the first term from the second one: \( 42 - 21 = 21 \).

The common difference tells you how much each step adds to the total, much like counting steps on a staircase where each step is of equal height. Knowing this difference not only helps in predicting future terms but also plays a vital role when using formulas to find specific terms in the progression. Always identify the common difference early as it is key to understanding the nature of the sequence.

Using the Arithmetic Sequence Formula

The arithmetic sequence formula is your ultimate tool for navigating arithmetic progressions. It's expressed as \( T_n = a + (n - 1) d \), where \( T_n \) is the \( n \)-th term you're looking for, \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

• For example, to find which term in the sequence equals 420, we use \( 420 = 21 + (n - 1)21 \).
• By simplifying, \( 420 = 21n \), and then solving for \( n \), you find \( n = 20 \).

This formula not only helps you find any term position but also verifies if a specific number is part of your sequence. It acts like a mathematical GPS, pinpointing exactly where you are in your progression journey. Always ensure to substitute correctly into this formula, as accurate substitution will lead you to the right answer.