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Chapter 14: Problem 75

Explain how to find the sum of the first \(n\) terms of an arithmetic sequence without having to add up all the terms.

Short Answer

Expert verified
The sum of the first \(n\) terms of an arithmetic sequence is given by the formula \(S_n = \frac{n}{2} (a_1 + a_n)\), where \(S_n\) is the sum, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the nth term. This formula allows finding the sum without going through the process of adding every term.

Step by step solution

01

Identify the Given

Identify the first term (\(a_1\)), the difference (\(d\)), and the number of terms (\(n\)) in the arithmetic sequence. If the nth term is not directly known, it can be found using the relation: \(a_n = a_1 + (n - 1) * d\).
02

Apply the Formula

Substitute the known values of \(a_1\), \(a_n\), and \(n\) in the formula \(S_n = \frac{n}{2} (a_1 + a_n)\) to find the sum of the first \(n\) terms.
03

Evaluate the Expression

After substituting the values, solve the expression to calculate \(S_n\), which is the sum of the first \(n\) terms of the arithmetic sequence.

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