Problem 66 - College Algebra

Question

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

Step-by-Step Solution

Answer

The sum of the first \(n\) terms of an arithmetic sequence can be found using the formula \(S = n/2 * (a + l)\) where \(n\) is the number of terms, \(a\) is the first term and \(l\) is the last term.

Step 1: Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. For instance, 1, 3, 5, 7, 9 is an arithmetic sequence with a common difference of 2.

Step 2: Recognizing the First and Last Terms

Let's denote the first term of the sequence as \(a\) and the nth term (the last term) as \(l\). In general, the nth term of an arithmetic sequence is given by \(l = a + (n-1)*d\) where \(d\) is the common difference.

Step 3: Deriving the Sum Formula

The sum \(S\) of the first \(n\) terms of an arithmetic sequence can be found without adding up the terms individually through the formula \(S = n/2 * (a + l)\), which is essentially the average of the first and last term, multiplied by the number of terms.

Step 4: Applying the Sum Formula

To find the sum of the first \(n\) terms, simply replace \(n\), \(a\), and \(l\) in the formula with their respective values and solve for \(S\).