Question

\(\mathrm{A}(\mathrm{n})\) ____________ is a sum of numbers that accumulates with each iteration of a loop.

Short answer

Question: Write down the generalized formula for the sum of an arithmetic sequence with n terms, a first term of a_1, and a common difference of c. Answer: The generalized formula for the sum of an arithmetic sequence with n terms, a first term of a_1, and a common difference of c is A(n) = (n/2) (2a_1 + (n - 1)c).

Step-by-step solution

Analyze the arithmetic sequence

We are provided with an arithmetic sequence with \(n\) terms. The first term is \(a_1\) and the common difference between consecutive terms is \(c\). The summation of the sequence, denoted as \(\mathrm{A}(\mathrm{n})\), needs to be found.

Express the nth term of the sequence

To express the \(n\)th term of the sequence, we can use the following formula: \(A_n = a_1 + (n - 1)c\) Here, \(A_n\) represents the \(n\)th term of the sequence, \(a_1\) is the first term, \(c\) is the common difference, and \(n\) is the number of terms.

Write down the summation formula of an arithmetic sequence

We can express the sum of the arithmetic sequence using the following summation formula: \(\mathrm{A}(\mathrm{n}) = \dfrac{n}{2} \left(a_1 + A_n\right)\) Here, \(\mathrm{A}(\mathrm{n})\) is the sum of the sequence, \(n\) is the number of terms, \(a_1\) is the first term, and \(A_n\) is the \(n\)th term.

Substitute the nth term expression into the summation formula

Now, substitute the expression for the \(n\)th term from Step 2 into the summation formula from Step 3: \(\mathrm{A}(\mathrm{n}) = \dfrac{n}{2} \left(a_1 + \left(a_1 + (n - 1)c\right)\right)\)

Simplify the summation formula

Finally, simplify the expression to obtain the sum of the arithmetic sequence, represented as \(\mathrm{A}(\mathrm{n})\): \(\mathrm{A}(\mathrm{n}) = \dfrac{n}{2} (2a_1 + (n - 1)c)\) This is the generalized formula for the sum of an arithmetic sequence with \(n\) terms, a first term of \(a_1\), and a common difference of \(c\).

Key concepts

summation formula

In arithmetic sequences, the summation formula is a valuable tool for calculating the sum of all the terms in the sequence quickly and efficiently. In general, an arithmetic sequence is defined by a first term, typically denoted as \(a_1\), and a constant difference between consecutive terms, known as the common difference \(c\). The summation formula helps us find the total sum of the numbers in the sequence without having to add each term manually, which can be very tedious.

To calculate the sum \(\mathrm{A}(\mathrm{n})\) of the first \(n\) terms of an arithmetic sequence, we use the following formula:

• \(\mathrm{A}(\mathrm{n}) = \dfrac{n}{2} (a_1 + A_n)\)

Here, \(n\) is the number of terms, \(a_1\) is the first term, and \(A_n\) is the \(n\)th term. Using this formula, you can quickly calculate the total sum by substituting these variables into the equation.

If you already know the \(n\)th term using the \(A_n = a_1 + (n - 1)c\) formula, calculating the sum becomes much more efficient, helping you finish arithmetic sequence problems with ease.

nth term

The concept of the \(n\)th term in an arithmetic sequence is essential for understanding how the sequence progresses and for finding any specific term without listing all previous terms. The formula for the \(n\)th term is:

• \(A_n = a_1 + (n - 1)c\)

This formula allows you to determine the value of a specific term in the sequence by knowing only the first term \(a_1\), the position in the sequence \(n\), and the common difference \(c\).

Using the \(n\)th term formula is practical in many scenarios, such as finding the last term when you know the number of terms and other key sequence characteristics. It also enables you to prepare for further calculations, such as summing the terms using the summation formula.

Mastering this concept is crucial, as it is the foundation for most operations involving arithmetic sequences. It equips you with a straightforward method to explore any term in the sequence directly.

common difference

The common difference in an arithmetic sequence is a fundamental characteristic that defines how the sequence progresses. It is symbolized by \(c\) and represents the uniform amount added to each term to obtain the next one. This property ensures that all consecutive terms in an arithmetic sequence are evenly spaced.

Understanding the common difference is pivotal because it allows you to determine other important sequence characteristics such as finding the \(n\)th term or calculating the sequence's total sum using the summation formula. The common difference can be found by subtracting any term in the sequence from its succeeding term.

• Mathematically, if you have two consecutive terms \(a_n\) and \(a_{n+1}\), \(c = a_{n+1} - a_n\)

Knowing the value of the common difference lets you easily build or predict any portion of the sequence as it stays constant throughout.

Grasping the concept of the common difference not only helps in solving arithmetic sequence problems but also in understanding patterns and relationships in numerical data.