Arithmetic Series
Deriving and applying the formula for the sum of the first n terms.
Introduction
An arithmetic series is the SUM of an arithmetic sequence's terms. Instead of adding term by term, we use Gauss's elegant formula: pair the first and last terms, multiply by the number of pairs.
Prerequisite Connection
You can find the n-th term using .
Today's Increment
We derive the sum formula .
Why This Matters
Arithmetic series calculate totals in finance, physics, and counting problems.
Key Concepts
Sum Formula (Version 1)
Average of first and last, times number of terms.
Sum Formula (Version 2)
Use when you know and but not .
Worked Examples
Example 1: Sum with Known Terms (Basic)
Find for 5, 8, 11, 14, ...
, ,
Example 2: Finding n from Sum (Intermediate)
How many terms of 3 + 7 + 11 + ... give sum 820?
→
terms
Example 3: Sum of Consecutive Integers (Advanced)
Find the sum of all integers from 50 to 150.
Identify: , ,
Number of terms:
Apply the formula:
Common Pitfalls
Using wrong value for n
Count the actual number of terms, not the last term value.
Forgetting to find
Formula 1 needs the last term. Calculate it first.
Real-World Application
Salary Over Time
Starting at $50,000 with $2,000 annual raises, 10-year total earnings:
Practice Quiz
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