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Chapter 5: Problem 25

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

Short Answer

Expert verified
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

01

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.
02

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.
03

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.
04

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: \(\mathrm{A}(\mathrm{n}) = \dfrac{n}{2} \left(a_1 + \left(a_1 + (n - 1)c\right)\right)\)
05

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 \(c\).

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