Problem 33
Question
Use the given information about the arithmetic sequence with common difference d to find a and a formula for \(a_{n}\). $$a_{4}=12, d=2$$
Step-by-Step Solution
Verified Answer
Question: Given an arithmetic sequence with the fourth term being 12 and a common difference of 2, find the first term and the general formula for the nth term.
Answer: The first term of the sequence is 6 and the general formula for the nth term is \(a_{n} = 6 + (n - 1)(2)\).
1Step 1: Write the formula for the 4th term
Let's first write down the formula for the fourth term in the sequence, \(a_4 = a + (4 - 1)d\).
2Step 2: Substitute the given information
Now we will substitute the given information, \(a_4 = 12\) and \(d = 2\), into the formula:
$$12 = a + (4 - 1)(2)$$
3Step 3: Solve for a
Now we solve the equation for a:
$$12 = a + (3)(2)$$
$$12 = a + 6$$
$$a = 12 - 6$$
$$a = 6$$
So the first term a is 6.
4Step 4: Write the formula for the nth term
Now that we have found the value of the first term (a = 6) and we have the common difference (d = 2), we can write the general formula for the nth term, \(a_n\):
$$a_{n} = 6 + (n - 1)(2)$$
This is the formula for the nth term of the given arithmetic sequence.
Key Concepts
Common DifferenceRecursive Formula for SequencesFinding the nth TermArithmetic Series
Common Difference
Understanding an arithmetic sequence requires grasping its core defining feature: the common difference. In the sequence, each term after the first is created by adding a constant amount to the previous term. This constant amount is known as the common difference, denoted by 'd'.
For instance, if a sequence goes 4, 6, 8, 10,..., the common difference 'd' is 2 since each term is 2 more than the term before it. Identifying the common difference is crucial for finding other terms in the sequence and establishing the pattern that defines the sequence's behavior. The common difference helps determine everything from specific terms to the overall structure of the sequence.
For instance, if a sequence goes 4, 6, 8, 10,..., the common difference 'd' is 2 since each term is 2 more than the term before it. Identifying the common difference is crucial for finding other terms in the sequence and establishing the pattern that defines the sequence's behavior. The common difference helps determine everything from specific terms to the overall structure of the sequence.
Recursive Formula for Sequences
A recursive formula for a sequence provides a way to determine each term using its preceding term(s). In terms of arithmetic sequences, the recursive formula often looks something like this:
\[a_{n} = a_{n-1} + d\]
where \(a_{n}\) is the nth term, \(a_{n-1}\) is the previous term, and 'd' is the common difference. The formula essentially states that to get any term in the sequence, take its preceding term and add the common difference. It is important because it builds the sequence one term at a time, demonstrating the underlying process of sequence generation.
\[a_{n} = a_{n-1} + d\]
where \(a_{n}\) is the nth term, \(a_{n-1}\) is the previous term, and 'd' is the common difference. The formula essentially states that to get any term in the sequence, take its preceding term and add the common difference. It is important because it builds the sequence one term at a time, demonstrating the underlying process of sequence generation.
Finding the nth Term
Finding the nth term of an arithmetic sequence is a fundamental task. It enables us to determine the value of any term in the sequence, regardless of its position. The formula to find the nth term of an arithmetic sequence is given by:
\[a_{n} = a + (n - 1)d\]
where \(a_{n}\) is the nth term, 'a' is the first term, 'n' is the term number, and 'd' is the common difference. This formula reflects the predictable nature of arithmetic sequences and lets us leap directly to any term without calculating all the previous terms. It's a powerful tool for analysis and has countless applications in problems involving sequences.
\[a_{n} = a + (n - 1)d\]
where \(a_{n}\) is the nth term, 'a' is the first term, 'n' is the term number, and 'd' is the common difference. This formula reflects the predictable nature of arithmetic sequences and lets us leap directly to any term without calculating all the previous terms. It's a powerful tool for analysis and has countless applications in problems involving sequences.
Arithmetic Series
When we talk about an arithmetic series, we refer to the sum of the terms of an arithmetic sequence. It's useful when we need to find the total of a sequence over a specified number of terms. The formula to calculate the sum of the first 'n' terms of an arithmetic series is:
\[S_{n} = \frac{n}{2}(2a + (n - 1)d)\]
or equivalently
\[S_{n} = \frac{n}{2}(a + a_{n})\]
where \(S_{n}\) is the sum of the first 'n' terms, 'a' is the first term, \(a_{n}\) is the nth term, and 'd' is the common difference. This formula comes in handy for solving real-world problems that require the sum of a series of numbers, such as financial calculations for annuities or understanding patterns in coding.
\[S_{n} = \frac{n}{2}(2a + (n - 1)d)\]
or equivalently
\[S_{n} = \frac{n}{2}(a + a_{n})\]
where \(S_{n}\) is the sum of the first 'n' terms, 'a' is the first term, \(a_{n}\) is the nth term, and 'd' is the common difference. This formula comes in handy for solving real-world problems that require the sum of a series of numbers, such as financial calculations for annuities or understanding patterns in coding.
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