Question

For the sequence \(2,5,8,11,14, \ldots\) (a) What is \(a_{4}\) ? (b) If \(a_{i}=5,\) what is \(i ?\) (c) If \(a_{n}=a_{n-1}+c,\) what is \(c ?\) (d) What is a possible formula for \(a_{n}\) ?

Short answer

Answer: The 4th term of the given arithmetic sequence is \(11\), the position of the term with value 5 is \(2\), the common difference between the terms is \(3\), and a possible formula for the nth term is \(a_{n} = 3n - 1\).

Step-by-step solution

Identify the sequence type

Observe that the successive terms in the given sequence are obtained by adding a fixed constant to the previous term: to go from \(2\) to \(5\) we add \(3\), from \(5\) to \(8\) we add \(3\) again, and so on. This indicates that the given sequence is an arithmetic sequence with a common difference of \(3\).

(a) Find the 4th term of the sequence - \(a_{4}\)

To find the 4th term \(a_{4}\) of an arithmetic sequence, we can add the common difference (\(c\)) to the third term (\(a_{3}\)), which can be found by adding the common difference (\(c\)) to the second term (\(a_{2}\)). Since we know \(a_{1} = 2\) and \(c = 3\), we can calculate \(a_{4}\) by adding the common difference to successive terms: \(a_{{4}} = a_{{3}} + c = (a_{{2}}+c) + c = ((a_{{1}}+c) + c) + c = (2+(3+3)+3 = 2 + 9 = 11\) So, the 4th term of the sequence is \(11\).

(b) For \(a_{i}=5\), find the term number - \(i\)

We know that the first term \(a_{1}=2\), the common difference \(c=3\), and we want to find the term position \(i\) for \(a_{i}=5\). We start with the first term and keep adding the common difference until we reach the term \(5\): \(a_{{1}}=2\), \(a_{{2}}=a_{{1}}+c=2+3=5\). So, the position \(i\) for the term \(5\) is \(2\), as \(a_{2}=5\).

(c) Find the common difference \(c\)

We have observed earlier that the given sequence is an arithmetic sequence with equal differences between successive terms called the common difference \(c\). We can find \(c\) by subtracting the first term \(a_{1}\) from the second term \(a_{2}\): \(c = a_{2} - a_{1}= 5 - 2 = 3\). So, the common difference \(c\) for this arithmetic sequence is \(3\).

(d) Find a possible formula for \(a_{n}\)

To find a general formula for the arithmetic sequence, we can use the following formula, where \(a_{1}\) is the first term, \(c\) is the common difference, and \(n\) is the term number: $$a_{n} = a_{1} + (n - 1)c$$ Since we know that the first term \(a_{1} = 2\), and the common difference \(c = 3\), we can plug in these values to get the general formula for our sequence: $$a_{n} = 2 + (n - 1)3 = 2 + 3n - 3 = 3n - 1$$ So, a possible formula for the nth term \(a_{n}\) in this sequence is \(3n - 1\).

Key concepts

Common Difference

In an arithmetic sequence, each number in the sequence is obtained by adding a fixed amount to the previous number. This fixed amount is called the common difference, denoted as \( c \). Identifying the common difference is crucial because it helps determine the behavior and pattern of the entire sequence.

To find the common difference in an arithmetic sequence, subtract any term from the term that follows it. For example, in the sequence \( 2, 5, 8, 11, 14, \ldots \), taking the second term \( 5 \) and subtracting the first term \( 2 \), we find \( c = 5 - 2 = 3 \). You can verify this by repeating the process for any pair of consecutive terms:

• \( 8 - 5 = 3 \)
• \( 11 - 8 = 3 \)

These equal differences confirm that every term is increased by the same common difference, \( c \).

Nth Term Formula

The nth term formula in an arithmetic sequence expresses any term based on its position in the sequence. It allows you to find the term at any given position \( n \) without needing to sequentially compute each prior term.

The general nth term formula for an arithmetic sequence is given by:
\[a_{n} = a_{1} + (n - 1)c\]
where:

• \( a_{n} \) is the nth term you're trying to find.
• \( a_{1} \) is the first term of the sequence.
• \( n \) is the position of the term.
• \( c \) is the common difference.

For our sequence \( 2, 5, 8, 11, 14, \ldots \), the first term \( a_{1} = 2 \) and common difference \( c = 3 \). Substituting these values, the formula becomes:
\[a_{n} = 2 + (n - 1) imes 3 = 3n - 1\]
This formula efficiently allows one to calculate any term directly, for instance, the 4th term is \( 3 \times 4 - 1 = 11 \).

Sequence Identification

Identifying the type of sequence you are dealing with is essential for using the correct formulas and properties. In arithmetic sequences, each term increases by a constant known as the common difference. Recognizing these patterns is crucial for predicting future terms and understanding the sequence.

To identify a sequence as arithmetic, examine the differences between successive terms. If they are consistent, the sequence is arithmetic. For example, consider the given sequence \( 2, 5, 8, 11, 14, \ldots \). Calculating the differences:

• \( 5 - 2 = 3 \)
• \( 8 - 5 = 3 \)
• \( 11 - 8 = 3 \)

Since these differences are all equal, you can confirm it's an arithmetic sequence. Once identified, you can employ formulas appropriate to arithmetic sequences, such as the nth term formula or finding specific terms using the common difference.