Problem 75 - Algebra and Trigonometry

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

Find the first term, last term and count of terms in the sequence. Then apply the sum formula \(S_n = \frac{n}{2}(a_1 + a_n)\) to calculate the sum of the first \(n\) terms.

Step 1: Identify the first and nth term

Before calculating the sum of the series, identify the first and nth term of the arithmetic series. These two values will be used in the formula to find the sum of the series.

Step 2: Identify the number of terms to be added

The 'n' in the formula represents the number of terms to be added in the arithmetic series. Identify this value and record it for use in the formula.

Step 3: Apply the Sum Formula

Once the first term, nth term, and total number of terms have been identified, insert these values into the arithmetic sum formula, \(S_n = \frac{n}{2}(a_1 + a_n)\), to calculate the sum of the first \(n\) terms of the sequence. The result is the sum of the first \(n\) terms without having to add all the individual terms.