Question

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{t_{n}\right\\}=\left\\{\frac{1}{2}-\frac{1}{3} n\right\\} $$

Short answer

The common difference is \( -\frac{1}{3} \). The first four terms are \( \frac{1}{6} \), \( -\frac{1}{6} \), \( -\frac{1}{2} \), and \( -\frac{5}{6} \).

Step-by-step solution

Define the General Formula

The general formula of the sequence is given by \( t_{n} = \frac{1}{2} - \frac{1}{3} n \). This formula tells us how to find the nth term of the sequence.

Find the First Term

To find the first term \( t_1 \), substitute \( n = 1 \) into the general formula: \[ t_1 = \frac{1}{2} - \frac{1}{3} \times 1 = \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

Find the Second Term

To find the second term \( t_2 \), substitute \( n = 2 \) into the general formula: \[ t_2 = \frac{1}{2} - \frac{1}{3} \times 2 = \frac{1}{2} - \frac{2}{3} = \frac{3}{6} - \frac{4}{6} = -\frac{1}{6} \]

Find the Third Term

To find the third term \( t_3 \), substitute \( n = 3 \) into the general formula: \[ t_3 = \frac{1}{2} - \frac{1}{3} \times 3 = \frac{1}{2} - 1 = \frac{3}{6} - \frac{6}{6} = -\frac{3}{6} = -\frac{1}{2} \]

Find the Fourth Term

To find the fourth term \( t_4 \), substitute \( n = 4 \) into the general formula: \[ t_4 = \frac{1}{2} - \frac{1}{3} \times 4 = \frac{1}{2} - \frac{4}{3} = \frac{3}{6} - \frac{8}{6} = -\frac{5}{6} \]

Identify the Common Difference

The common difference \( d \) in an arithmetic sequence is found by subtracting the first term from the second term: \[ d = t_2 - t_1 = -\frac{1}{6} - \frac{1}{6} = -\frac{2}{6} = -\frac{1}{3} \]. Confirmation: \[ t_3 - t_2 = -\frac{1}{2} - ( -\frac{1}{6} ) = -\frac{3}{6} + \frac{1}{6} = -\frac{2}{6} = -\frac{1}{3} \], \[ t_4 - t_3 = -\frac{5}{6} - ( -\frac{1}{2} ) = -\frac{5}{6} + \frac{3}{6} = -\frac{2}{6} = -\frac{1}{3} \]

Key concepts

Common Difference

Understanding the concept of 'common difference' is essential to identify and work with arithmetic sequences. In arithmetic sequences, the common difference, denoted by \(d\), is the constant amount that you add or subtract from one term to get to the next term.
To find the common difference in the given sequence \(t_n = \frac{1}{2} - \frac{1}{3} n\), you can subtract the first term from the second term:
\[d = t_2 - t_1\]
Based on the solution:
\[d = -\frac{1}{6} - \frac{1}{6} = -\frac{2}{6} = -\frac{1}{3}\]
This common difference \(-\frac{1}{3}\) is consistent for each pair of successive terms in this sequence, confirming it is arithmetic.

Nth Term Formula

The nth term formula helps us to find any term in an arithmetic sequence without needing to list all previous terms. For the sequence \(t_n = \frac{1}{2} - \frac{1}{3} n\), this formula shows how the terms are generated. Each term in this specific sequence is calculated by:
\[t_n = \frac{1}{2} - \frac{1}{3} n\]
This means that to find any term \(t_n\), substitute the position number \(n\) into the formula. For example, to find the 5th term (\(t_5\)):
\[t_5 = \frac{1}{2} - \frac{1}{3} \times 5 = \frac{1}{2} - \frac{5}{3} = \frac{3}{6} - \frac{10}{6} = -\frac{7}{6}\]
With the nth term formula, you can directly calculate any term in the sequence.

Sequence Identification

Identifying whether a sequence is arithmetic involves checking if there is a constant common difference between consecutive terms. For the sequence \(t_n = \frac{1}{2} - \frac{1}{3} n\), through the derived values:

• \(t_1 = \frac{1}{6}\)
• \(t_2 = -\frac{1}{6}\)
• \(t_3 = -\frac{1}{2}\)
• \(t_4 = -\frac{5}{6}\)

Subtracting each term from the next confirms a common difference \(-\frac{1}{3}\), thus proving the sequence is arithmetic. This consistency is key to sequence identification.

Term Calculation

Calculating each term in an arithmetic sequence like \(t_n = \frac{1}{2} - \frac{1}{3} n\) requires substituting the desired position into the formula. Let's show how to find the first four terms:
1. For the first term \(t_1\):
\[t_1 = \frac{1}{2} - \frac{1}{3} \times 1 = \frac{1}{6}\]
2. For the second term \(t_2\):
\[t_2 = \frac{1}{2} - \frac{1}{3} \times 2 = -\frac{1}{6}\]
3. For the third term \(t_3\):
\[t_3 = \frac{1}{2} - \frac{1}{3} \times 3 = -\frac{1}{2}\]
4. For the fourth term \(t_4\):
\[t_4 = \frac{1}{2} - \frac{1}{3} \times 4 = -\frac{5}{6}\]
Understanding these calculations ensures you can correctly generate terms in any arithmetic sequence.