Problem 22 - Algebra 2

Question

Write an explicit formula for each sequence. Then find \(a_{12}\) $$ -2 \frac{1}{2},-2,-1 \frac{1}{2},-1, \ldots $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is given by \(a_n = -2 + (n-1) \cdot 0.5\) and the 12th term of the sequence is 3.5.

Step 1: Determine the first term and the common difference of the sequence

The first term \(a_1\) of the sequence is -2. The common difference \(d\) is obtained by subtracting any term from the term that follows it, for example: \(-2 - (-2.5) = 0.5\). Thus, the first term \(a_1\) is -2 and the common difference \(d\) is 0.5.

Step 2: Write down the explicit formula of the sequence

The sequence is an arithmetic sequence. The formula for the nth term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \cdot d\). Substituting \(a_1 = -2\) and \(d = 0.5\) into the equation, gives: \(a_n = -2 + (n-1) \cdot 0.5\).

Step 3: Find the 12th term of the sequence

We can find the 12th term \(a_{12}\) by substituting \(n = 12\) into the explicit formula. This gives: \(a_{12} = -2 + (12-1) \cdot 0.5 = -2 + 5.5 = 3.5\).

Key Concepts

Explicit Formula

An **explicit formula** provides a way to find any term in a sequence without having to calculate the previous terms. This type of formula is really handy because you can jump straight to the term you're interested in.
For arithmetic sequences, the explicit formula is defined as:\[ a_n = a_1 + (n-1) \cdot d \]where:

• \( a_n \) is the nth term
• \( a_1 \) is the first term of the sequence
• \( n \) represents the term number
• \( d \) is the common difference between the terms

In the sequence you provided, the first term \( a_1 \) is -2 and the common difference \( d \) is 0.5. Substituting these values into the explicit formula gives:\[ a_n = -2 + (n-1) \cdot 0.5 \]This equation allows you to find any term in the sequence by plugging in the desired term number for \( n \).

Common Difference

The **common difference** in an arithmetic sequence is the constant amount that you add or subtract to go from one term to the next. Understanding how to find the common difference is essential because it's a critical component of the explicit formula.
To calculate the common difference \( d \):

• Pick any two consecutive terms from the sequence
• Subtract the first term from the second term

For the sequence \(-2.5, -2, -1.5, -1, \ldots\), if you take the first two terms, you calculate:\[ d = (-2) - (-2.5) = 0.5 \]This calculation shows that we add 0.5 to each term to get the next term. The constant difference of 0.5 is what makes this sequence arithmetic, as it clearly defines the pattern of the sequence.

Nth Term Calculation

**Calculating the nth term** is straightforward once you have the explicit formula. It's all about plugging in the right value for \( n \) into the equation and solving it. This technique lets you find any term in the sequence without listing all the previous terms.
For the given sequence, we already know the explicit formula:\[ a_n = -2 + (n-1) \cdot 0.5 \]To find the 12th term, set \( n = 12 \) and substitute:\[ a_{12} = -2 + (12-1) \cdot 0.5 \]Simplifying inside the parentheses gives:\[ a_{12} = -2 + 11 \cdot 0.5 \]And simplifying further results in:\[ a_{12} = -2 + 5.5 = 3.5 \]This final calculation shows that the 12th term of the sequence is 3.5, demonstrating how quickly and easily the explicit formula can get you right to the term you need.