Problem 26 - Algebra and Trigonometry

Question

Find the indicated term in each arithmetic sequence. $$ \text { 80th term of }-1,1,3, \ldots $$

Step-by-Step Solution

Answer

157

Step 1: Identify initial term and common difference

Identify the first term () and the common difference () in the arithmetic sequence. The first term () is (-1) and the common difference () can be calculated as follows: ( = 1 - (-1) = 2).

Step 2: Use the nth-term formula of an arithmetic sequence

Apply the nth-term formula of an arithmetic sequence, which is given by:a_n = a_1 + (n - 1)dwhere a_n is the nth term, a_1 is the first term, d is the common difference, and n is the term number.

Step 3: Substitute the values

Substitute the known values into the formula: a_80 = -1 + (80 - 1) × 2This simplifies to:a_80 = -1 + 79 × 2.

Step 4: Perform the calculations

Calculate the result:a_80 = -1 + 158a_80 = 157.

Key Concepts

nth-term formula

In an arithmetic sequence, we use a special formula to find any term in the sequence. This is called the nth-term formula. The nth-term formula is given by: \( a_n = a_1 + (n - 1)d \). Here, \( a_n \) is the term you want to find, \( a_1 \) is the first term in the sequence, and \( d \) is the common difference. The variable \( n \) represents the position of the term within the sequence.
For example, if we want to find the 80th term in the sequence -1, 1, 3, ..., we need to identify \( a_1 \), \( d \), and \( n \).
Let's proceed to understand what these terms mean.

common difference

The common difference is the amount that each term in an arithmetic sequence increases or decreases by. We can find it by subtracting any term from the term that follows it.
For instance, in the sequence -1, 1, 3, ..., we can see that the common difference \( d \) is obtained as follows: \( 1 - (-1) = 2 \). Hence, \( d = 2 \).
The common difference helps in constructing the entire sequence and is crucial for using the nth-term formula. A positive common difference indicates an increasing sequence, while a negative common difference indicates a decreasing sequence.

initial term

The initial term, also referred to as the first term, is the starting point of your sequence. In our example, this is given as the first number in the list. For the sequence -1, 1, 3, ..., the initial term \( a_1 \) is -1.
Knowing the initial term is essential because it sets the starting point for applying the nth-term formula. Every subsequent term in the sequence builds off this first term by repeatedly applying the common difference.

sequence calculation

Now, let's put all the pieces together and calculate the desired term in our sequence. For the problem given, we want to find the 80th term of the sequence -1, 1, 3, .... Using the nth-term formula: \( a_n = a_1 + (n - 1)d \), we substitute the values we have identified:
- Initial term \( a_1 \) = -1- Common difference \( d \) = 2- Term number \( n \) = 80
Substituting these values, we get:
\( a_{80} = -1 + (80 - 1) \times 2 \)
Now calculate it step-by-step:
\( a_{80} = -1 + 79 \times 2 \)
\( a_{80} = -1 + 158 \)
\( a_{80} = 157 \)
Thus, the 80th term of the sequence is 157.
By understanding and applying the nth-term formula along with the given initial term and common difference, you can easily find any term in an arithmetic sequence.