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Chapter 7: Problem 35

Restate the symbolic version of the formula for evaluating an arithmetic series using summation notation.

Short Answer

Expert verified
The symbolic version of the formula for evaluating an arithmetic series using summation notation is: S_n = \(\sum_{i=1}^{n} (a_1 + (i-1) * d)\)

Step by step solution

01

Understand the Arithmetic Series

An arithmetic series is the sum of terms of an arithmetic sequence that has a common difference, d, between successive terms. We can represent the consecutive terms of this sequence as: a_1, a_2, a_3, ..., a_n, where a_1 is the first term, and a_n is the last term.
02

Derive the Arithmetic Series Formula

To derive the formula, let's consider the sum of an arithmetic series, S_n, which consists of n terms, i.e., S_n = a_1 + a_2 + a_3 + ... + a_n. Let's express the series in the reversed order and add the terms: S_n = a_n + a_{n-1} + a_{n-2} + ... + a_1 S_n + S_n = (a_1 + a_n) + (a_2 + a_{n-1}) + (a_3 + a_{n-2}) + ... + (a_n + a_1) Since there are n pairs of terms and each pair sum is equal to (a_1 + a_n), we have the following: 2 * S_n = n * (a_1 + a_n) Solving for S_n, we get the formula for the sum of an arithmetic series: S_n = (n * (a_1 + a_n)) / 2
03

Represent the Formula with Summation Notation

Now we'll use the summation notation to represent the formula we derived in the previous step. We can rewrite the arithmetic series S_n as the sum of each term of the arithmetic sequence using this notation: S_n = \(\sum_{i=1}^{n} a_i\) where \(a_i = a_1 + (i-1) * d\), which is the general term for an arithmetic sequence. After substituting \(a_i\) in the summation notation, we get: S_n = \(\sum_{i=1}^{n} (a_1 + (i-1) * d)\)

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Most popular questions from this chapter

Explain how the formula $$ e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots $$ leads to the approximation \(e^{x} \approx 1+x\) if \(x\) is very close to 0 (which we derived by another method in Section 4.4).

Find the tenth term of a geometric sequence whose second term is 3 and whose seventh term is \(11 .\)

Give the first four terms of the specified recursive sequence. \(a_{1}=2, a_{2}=3,\) and \(a_{n+2}=a_{n} a_{n+1}\) for \(n \geq 1\).

Find the fifth term of an arithmetic sequence whose second term is 8 and whose third term is 14.

In Exercises \(1-10,\) evaluate the arithmetic series. \(300+293+286+\cdots+55+48+41\)
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