Problem 75 - 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 \(S_n = n/2 * (a + l)\), where \(S_n\) is the sum, \(n\) is number of terms, \(a\) is the first term and \(l\) is the last term. The first term \(a\) and last term \(l = a +(n-1)d\) can be easily identified in the sequence and substituted in the formula to find the sum.

Step 1: Understand the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This simply means that the numbers progress in a linear or straight-line manner. For example, the sequence '1, 3, 5, 7' is an arithmetic sequence with a common difference of 2.

Step 2: Identify the Elements in the Formula

Before using the formula, it's necessary to identify the values in the given sequence. The first term \(a\) is just the first number in the sequence and the last term \(l\) will be the nth term in the sequence, which is \(a + (n-1)d\) where \(d\) is the common difference.

Step 3: Use the Sum Formula

Once all the elements are identified, substitute the values of \(n\), \(a\) and \(l\) in the sum formula \(S_n = n/2 * (a + l)\). This will give the sum of the first \(n\) terms of an arithmetic sequence without having to add up all the terms individually.