Problem 18 - Algebra 2

Question

Write an explicit formula for each sequence. Then find \(a_{12}\) $$ 4,5,6,7,8, \dots $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is \(a_n = n + 3\). The 12th term of the sequence is 15.

Step 1: Identify the common difference

The common difference of an arithmetic sequence is the difference between any two consecutive terms. For the given sequence \(4,5,6,7,8, \dots \), the common difference is found out by subtracting the first term from the second term, or the second from the third, and so on. Here, the common difference (\(d\)) is \(5 - 4 = 1\).

Step 2: Formulate the explicit formula

The general form of an arithmetic sequence is \(a_n = a_1 + (n - 1) * d\), where \(a_n\) is the nth term, \(a_1\) is the first term, n is the term number, and d is the common difference. In this case, since the first term (\(a_1\)) is 4 and the common difference (\(d\)) is 1, the explicit formula for the sequence is \(a_n = 4 + (n - 1) * 1\). This simplifies to \(a_n = n + 3\).

Step 3: Find \(a_{12}\)

Substituting \(n = 12\) into the formula \(a_n = n + 3\), we find \(a_{12} = 12 + 3 = 15\).

Key Concepts

Explicit Formula

An explicit formula is a way to find any term in a sequence without listing all previous terms. It's like having a shortcut to the 100th step in a staircase without having to count every step along the way.
For arithmetic sequences, the explicit formula is written as:

• \( a_n = a_1 + (n - 1) \cdot d \)

Here:

• \(a_n\) is the nth term you're looking for.
• \(a_1\) is the first term of the sequence.
• \(n\) is the term number.
• \(d\) is the common difference, which you'll learn more about soon.

In the given sequence, the first term \(a_1\) is 4, and the common difference \(d\) is 1. Plugging these into the formula gives us:

• \(a_n = 4 + (n - 1) \cdot 1\)

This simplifies to \(a_n = n + 3\), allowing us to find any term just by substituting the value of \(n\). For \(n = 12\), we get \(a_{12} = 15\). It's a very efficient way to jump ahead in a sequence!

Common Difference

In an arithmetic sequence, each term is generated by adding a constant value, known as the "common difference," to the previous term. It's consistent and predictable, defining the sequence's uniform pattern.
To find the common difference \(d\) in a sequence like 4, 5, 6, 7, 8, ..., you simply subtract one term from the next.
This is how:

• Take the second term (5) and subtract the first term (4): \(5 - 4 = 1\).
• Repeat with other terms to double-check: \(6 - 5 = 1\).

The common difference is 1, meaning each term is 1 more than the one before.
This constant \(d\) is crucial because it allows the explicit formula to function, linking one term to the next seamlessly.

Arithmetic Progression

An arithmetic progression is a sequence of numbers where each number after the first is obtained by adding a constant difference.
This makes the sequence linear and easy to predict. The general look of such a sequence is:

• \( a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \ldots \)

This repetitive pattern is why arithmetic sequences are often used in real-life situations, such as calculating loan payments or scheduling events.
For the sequence 4, 5, 6, 7, 8, ..., each term is simply the previous term plus 1, starting from the first term 4.
With the explicit formula \(a_n = n + 3\), any future term can be predicted, confirming the sequence's structure and reliability.
Understanding arithmetic progressions provides a strong foundation for exploring more complex mathematical concepts.