Partial sums of arithmetic series

Learning Objectives Define an arithmetic series and its partial sum (Sₙ). Identify the first term (a₁), the common difference (d), and the number of terms (n) for a given arithmetic series. State and apply the two primary formulas for calculating the partial sum of an arithmetic series. Calculate the sum of a finite number of terms in an arithmetic series. Solve for a missing variable (such as n, a₁, or d) when the partial sum is known. Model and solve real-world problems involving partial sums of arithmetic series. Imagine stacking cans in a pyramid at the grocery store. How could you quickly calculate the total number of cans without counting them one by one? 🥫 This tutorial introduces the concept of a partial sum of an arithmetic series, a powerful tool for finding the...

TermDefinitionExample Arithmetic SequenceAn ordered list of numbers where the difference between any two consecutive terms is constant.The sequence 5, 8, 11, 14, 17, ... is an arithmetic sequence because you add 3 to get to the next term. Common Difference (d)The constant value added to each term to get the next term in an arithmetic sequence.In the sequence 5, 8, 11, 14, ..., the common difference (d) is 3. SeriesThe sum of the terms in a sequence.For the sequence 2, 4, 6, 8, the corresponding series is 2 + 4 + 6 + 8. Arithmetic SeriesThe sum of the terms of an arithmetic sequence.The sum 5 + 8 + 11 + 14 + 17 is an arithmetic series. Partial Sum (Sₙ)The sum of a specified number of terms from the beginning of a sequence. It is denoted by Sₙ, where 'n' is the number of terms bei...

Partial Sum Formula (using first and last term) S_n = \frac{n}{2}(a_1 + a_n) Use this formula when you know the number of terms (n), the first term (a₁), and the last term (aₙ). It is the most direct way to find the sum. Partial Sum Formula (using first term and common difference) S_n = \frac{n}{2}[2a_1 + (n-1)d] Use this formula when you know the number of terms (n), the first term (a₁), and the common difference (d), but you do not know the last term (aₙ). Nth Term Formula (for finding aₙ) a_n = a_1 + (n-1)d This is the formula for a term in an arithmetic sequence. It is often used to find the last term (aₙ) needed for the first partial sum formula.