Question

Find the indicated term in each arithmetic sequence. $$ \text { 80th term of } 2, \frac{5}{2}, 3, \frac{7}{2}, \ldots $$

Short answer

The 80th term is \(\frac{83}{2}\).

Step-by-step solution

Identify the first term (\(a\))

The first term of the sequence is given as \(a = 2\).

Find the common difference (\(d\))

To find the common difference, subtract the first term from the second term: \[ d = \frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2} \]

Use the formula for the nth term

The formula for finding the nth term of an arithmetic sequence is: \[ a_n = a + (n-1)d \]

Substitute the known values into the formula

Substitute \(a = 2\), \(d = \frac{1}{2}\), and \(n = 80\) into the formula: \[ a_{80} = 2 + (80-1)\frac{1}{2} \]

Simplify the expression

First, calculate the term inside the parentheses: \[ 80 - 1 = 79 \]Then, multiply by the common difference: \[ 79 \times \frac{1}{2} = \frac{79}{2} \]Finally, add this to the first term: \[ a_{80} = 2 + \frac{79}{2} = \frac{4}{2} + \frac{79}{2} = \frac{83}{2} \]

Key concepts

first term

The first term of an arithmetic sequence is the starting point of the sequence. In any list of numbers forming an arithmetic progression, the first term is denoted by the letter \(a\). For example, in the sequence \(2, \frac{5}{2}, 3, \frac{7}{2}, \text{...}\), the first term is \(2\). This sets the stage for the entire sequence, providing the baseline from which every other term is calculated.

common difference

The common difference in an arithmetic sequence is the amount by which each term increases (or decreases) from the previous term. It is denoted by the letter \(d\). To find the common difference, simply subtract the first term from the second term. For instance, in our sequence \(2, \frac{5}{2}, 3, \frac{7}{2}, \text{...}\), the common difference is:
\( d = \frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2} \)
This step is crucial because the value of \(d\) determines the spacing between each number in the sequence.

• If \(d\) is positive, the sequence will increase.
• If \(d\) is negative, the sequence will decrease.
• If \(d\) is zero, the sequence will be constant.

nth term formula

To find any term in an arithmetic sequence, you can use the nth term formula. This formula is:
\[a_n = a + (n-1)d\]
where:

• \(a_n\) is the nth term you want to find.
• \(a\) is the first term.
• \(d\) is the common difference.
• \(n\) is the term's position in the sequence (1st, 2nd, 3rd, etc.).

For example, to find the 80th term of the sequence \(2, \frac{5}{2}, 3, \frac{7}{2}, \text{...}\), you substitute the known values into the formula:
\( a_{80} = 2 + (80-1)\frac{1}{2} \)
Simplify it step by step to get:
\( 79 \times \frac{1}{2} = \frac{79}{2} \)
Then, add this to the first term:
\[a_{80} = 2 + \frac{79}{2} = \frac{4}{2} + \frac{79}{2} = \frac{83}{2}\]

arithmetic progression

An arithmetic progression (or arithmetic sequence) is a list of numbers in which each term is obtained by adding a constant difference to the previous term. This concept is fundamental in various areas of math and science.
Here are some key points to remember about arithmetic progression:

• Each term is formed by adding the common difference \(d\) to the previous term.
• It's represented as \(a, a+d, a+2d, a+3d, \text{...}\).
• The nth term formula \(a_n = a + (n-1)d\) helps in finding any term in the sequence.

Understanding arithmetic progressions is essential for solving various mathematical problems, including those involving sequences and series. It also forms the basis for more complex topics like geometric and harmonic sequences.