Master the art of summing arithmetic sequences using powerful formulas and clever techniques
You've learned about arithmetic sequences—now it's time to add them up! A finite arithmetic series is the sum of the terms in an arithmetic sequence.
For example, if you have the sequence 2, 5, 8, 11, 14, the corresponding series is:
But what if you need to add 100 terms? Or 1,000? Adding them one by one would take forever. That's where sum formulas become indispensable.
Legend has it that young Carl Friedrich Gauss amazed his teacher by quickly summing 1 + 2 + 3 + ... + 100. His method? Write the sum twice—once ascending, once descending—and add corresponding pairs.
This pairing trick works for any arithmetic sequence because the common difference ensures each pair has the same sum.
From Gauss's method, we derive two equivalent formulas for the sum of a finite arithmetic series:
Experiment with different sequences and see both formulas in action. Notice how they always produce the same result!
Watch how the sum accumulates as you add more terms. Each new term builds on the previous total.
Test your understanding with these practice problems, organized by difficulty level.
Now that you've mastered finite arithmetic series, you're ready to explore geometric series, where terms are multiplied by a constant ratio instead of adding a constant difference!