Problem 17 - Algebra 2

Question

Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=100, r=-20 $$

Step-by-Step Solution

Answer

The explicit formula for our sequence is \(a_{n}= 100+(n-1)*-20\). Using this, the first five terms of the sequence are 100, 80, 60, 40, 20.

Step 1: Identify the Form of the Sequence

The given sequence is an arithmetic sequence, where each term is calculated as \(a_{n}= a_{1}+(n-1)*d\). Here, \(a_{1}\) denotes the first term and \(d\) the common difference.

Step 2: Apply the Form of the Sequence to the Given Values

Plugging in the given values into our formula, so our specific term formula becomes \(a_{n}= 100+(n-1)*-20\). The term \(n-1\) generates the order number of the difference to be added to the original term. Note that since our common difference is negative, our sequence will decrease each time.

Step 3: Generate the First Five Terms

We substitute the values of \(n\) from 1 to 5 into the formula to get the first five terms. They are calculated as follows: \(a_{1} = 100+(1-1)*-20 = 100\), \(a_{2} = 100+(2-1)*-20 = 80\), \(a_{3} = 100+(3-1)*-20 = 60\), \(a_{4} = 100 + (4-1)*-20 = 40\), \(a_{5} = 100 + (5-1)*-20 = 20\).

Key Concepts

Explicit formula for sequences

In an arithmetic sequence, the explicit formula is a powerful tool. It allows us to find any term in the sequence without listing all the previous ones. The explicit formula for an arithmetic sequence is generally given by:

• \( a_n = a_1 + (n-1)d \)

• \( a_n \) represents the term you want to find.
• \( a_1 \) is the first term in the sequence.
• \( d \) is the common difference between the terms.
• \( n \) is the position of the term in the sequence.

When you plug the specific values from your exercise into this formula, you personalize it to predict any term of the sequence. For example, with \( a_1 = 100 \) and \( d = -20 \), the formula becomes \( a_n = 100 + (n-1)(-20) \). This formula can now be used to find terms beyond the first five or any term in case of large sequences.

Common difference

The common difference is a defining feature of an arithmetic sequence. It is the constant amount that you add (or subtract if negative) to get from one term in the sequence to the next.

• If the common difference is positive, the sequence increases with each term.
• If the common difference is negative, like in the provided exercise, the sequence decreases with each step.

In the given exercise, the common difference provided is \(-20\). This means each subsequent term is 20 less than the previous one.
This constant difference forms a linear pattern. Recognizing it helps you construct sequences quickly and efficiently without recalculating every single term from the start.

Generating terms of a sequence

After formulating the explicit formula, you can easily generate terms of the sequence.

• First, you set \( n \) to the position number of the desired term.
• Substitute \( n \) into the explicit formula.

For our specific formula, \( a_n = 100 + (n-1)(-20) \), generating the first five terms is straightforward:

• For \( n = 1 \), the term is calculated as 100.
• For \( n = 2 \), it is 80.
• For \( n = 3 \), it becomes 60.
• For \( n = 4 \), the result is 40.
• For \( n = 5 \), you find 20.

Each step represents a clear application of the explicit formula, demonstrating how useful it is for efficiently computing sequence terms. This is particularly helpful in academic scenarios where quick calculations preserve time and focus on problem-solving elements.