Change the first term and common difference (or common ratio) with the sliders, and see for yourself why the sum formula looks the way it does.
A sequence is a string of numbers lined up according to a fixed rule. The two most basic ones are: an arithmetic sequence, made by adding the same number every time, and a geometric sequence, made by multiplying by the same number every time. Whether the rule is "add" or "multiply" is the one difference that completely changes how a sequence looks.
What's remarkable about the sum formula for an arithmetic sequence is that the calculation finishes instantly no matter how many terms there are. Add the first term and the last term, multiply by the number of terms, and divide by two โ that's it. For a geometric sequence, if the common ratio is greater than 1, the value explodes as it goes on; if it's less than 1, it gets closer and closer to 0.
Use the sliders to change the first term and the common difference (or common ratio), and see for yourself how the bars and the area drawn by the sequence line up exactly with the sum formula.
According to legend, Gauss added up 1 through 100 in an instant. Flip the bars upside down and stack a second copy on top, and every pair adds up to the same value (first term + last term), so the whole thing can be calculated in a flash.
Adjust the common ratio (r) and see how quickly (or slowly) the bars grow. If the common ratio is greater than 1, it grows explosively; if it's less than 1 (between 0 and 1), it gets closer and closer to 0.
Enter a savings amount and interest rate yourself, and see how much it becomes after 20 years using a geometric sequence. Because it's "compound interest" โ interest that earns interest on top of interest already earned โ the graph gets steeper and steeper over time.