Question

In Problems 22-25, find the first four terms of the sequence and a formula for the general term. $$ a_{n}=a_{n-1}+4 ; a_{1}=2 $$

Short answer

Answer: The first four terms of the sequence are 2, 6, 10, and 14. The formula for the general term is \(a_n = 4n - 2\).

Step-by-step solution

Find the second term (a_{2})

Using the recursive formula \(a_{n}=a_{n-1}+4\), we can find the second term by substituting \(n=2\): $$ a_{2} = a_{2-1} + 4 = a_{1} + 4 = 2 + 4 = 6 $$ So, the second term of the sequence (\(a_{2}\)) is 6.

Find the third term (a_{3})

Using the recursive formula \(a_{n}=a_{n-1}+4\), we can find the third term by substituting \(n=3\): $$ a_{3} = a_{3-1} + 4 = a_{2} + 4 = 6 + 4 = 10 $$ So, the third term of the sequence (\(a_{3}\)) is 10.

Find the fourth term (a_{4})

Using the recursive formula \(a_{n}=a_{n-1}+4\), we can find the fourth term by substituting \(n=4\): $$ a_{4} = a_{4-1} + 4 = a_{3} + 4 = 10 + 4 = 14 $$ So, the fourth term of the sequence (\(a_{4}\)) is 14.

Determine the general formula for the sequence

Now that we have the first four terms of the sequence: 2, 6, 10, and 14, we can observe that the sequence is an arithmetic sequence, with a common difference of 4. The general formula for an arithmetic sequence is given by: $$ a_n = a_1 + (n - 1)d $$ Where \(a_n\) is the nth term, \(a_1\) is the first term, n is the position of the term, and d is the common difference. In our case, \(a_1 = 2\) and \(d=4\). Therefore, the general formula for this sequence is: $$ a_n = 2 + (n - 1)4 $$ We can further simplify this formula: $$ a_n = 2 + 4n - 4 = 4n - 2 $$ So, the general formula for the given sequence is \(a_n = 4n - 2\).

Key concepts

Recursive Formula

A recursive formula is a powerful tool in mathematics to define sequences. It gives us the next term in the sequence based on the current term. Instead of providing a direct expression for each term's value, a recursive formula lets us calculate the term step by step.
For example, in the given problem, the recursive formula is \( a_{n} = a_{n-1} + 4 \). This tells us that to find the \( n^{th} \) term, we take the \((n-1)^{th}\) term and simply add 4. This constant addition indicates that our sequence is arithmetic, as each term steadily builds upon the previous one.

• Start with a given term, like \( a_1 = 2 \).
• Use the recursive rule to generate subsequent terms.
• This stepwise computation shows the growth pattern of the sequence.

By following a recursive path, you gain a clear understanding of how each term is derived, making it easier to see how arithmetic sequences progress over time.

General Term Formula

While a recursive formula works like a road map from one term to the next, a general term formula allows us to jump directly to any term in the sequence without computing all the previous terms first.
In the context of our exercise, the sequence progresses uniformly by adding 4 each time, and the general term formula captures this pattern. It is derived by recognizing that arithmetic sequences follow a linear pattern. For our sequence, the formula is found to be \( a_n = 4n - 2 \).

• It enables calculating any term directly by substituting \( n \) into a formula.
• For \( n = 1 \), \( a_n = 4(1) - 2 = 2 \), verifying our starting term.
• For \( n = 2 \), \( a_n = 4(2) - 2 = 6 \), matching our earlier result.

The beauty of the general term formula lies in its ability to provide instant access to any term's value, just by plugging in the desired \( n \). This makes it invaluable for identifying terms far along in a sequence without tedious calculations.

Common Difference

The common difference is a hallmark feature of arithmetic sequences. It refers to the constant amount separating one term from the next in the series. Identifying this is crucial for forming both the recursive and general term formulas.
In the exercise you've worked through, the common difference is 4. Each term is 4 units more than the previous term, hence the formula \( a_n = a_{n-1} + 4 \).

• A key marker of arithmetic growth patterns.
• For our sequence, \( 6 - 2 = 4 \), \( 10 - 6 = 4 \), showing consistent separation.
• Directly influences the slope or coefficient in the general term formula, \( 4n \) signifies that increase.

Recognizing and calculating the common difference assists you not only in understanding arithmetic sequences but also in predicting how long such a sequence will continue in an orderly fashion. Comprehending this concept is fundamental to exploring further complex sequences and series.