Section 1
Pairing Terms Strategy for Arithmetic Series
Property
In an arithmetic series, terms equidistant from the ends have the same sum:
This allows pairing terms to find the sum efficiently.
In this Grade 4 AMC Math lesson from AoPS Introduction to Algebra, students learn how to find the sum of an arithmetic series using the formula: number of terms multiplied by the average of the first and last terms, expressed as n[2a + (n−1)d]/2. The lesson walks through Gauss's classic method of pairing terms, and covers special cases including the sum of the first n positive integers (n(n+1)/2) and the sum of the first n odd integers (n²). Students also practice setting up algebraic expressions to solve multi-step arithmetic series problems from the AMC 8 and AMC 10.
Section 1
Pairing Terms Strategy for Arithmetic Series
In an arithmetic series, terms equidistant from the ends have the same sum:
This allows pairing terms to find the sum efficiently.
Section 2
Sum of an Arithmetic Sequence
The sum, , of the first terms of an arithmetic sequence is
where is the first term and is the th term.
Section 3
Alternative Formula for Arithmetic Series Sum
The sum of an arithmetic series can be calculated using the alternative formula:
where is the number of terms, is the first term, and is the common difference.
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Section 1
Pairing Terms Strategy for Arithmetic Series
In an arithmetic series, terms equidistant from the ends have the same sum:
This allows pairing terms to find the sum efficiently.
Section 2
Sum of an Arithmetic Sequence
The sum, , of the first terms of an arithmetic sequence is
where is the first term and is the th term.
Section 3
Alternative Formula for Arithmetic Series Sum
The sum of an arithmetic series can be calculated using the alternative formula:
where is the number of terms, is the first term, and is the common difference.
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter