A list of numbers arranged according to a fixed rule is called a sequence. There are two basic types: an arithmetic sequence, made by adding the same number every time, and a geometric sequence, made by multiplying by the same number every time. Understand the difference between an "adding rule" and a "multiplying rule," and you've already conquered half of sequences.
There's a famous legend attached to arithmetic sequences. As a young student, Gauss was assigned by his teacher to add up the numbers from 1 to 100 as a punishment — and he came up with the answer, 5050, in just a few seconds. His secret was a trick of pairing up numbers with a flipped copy. 1 plus 100 is 101, 2 plus 99 is also 101, 3 plus 98 is also 101... pair the numbers this way and you always get 101, and since there are 50 such pairs, 101×50=5050 comes out in an instant.
Applying this exact trick to a general arithmetic sequence is exactly the sum formula. For an arithmetic sequence with first term a₁, last term aₙ, and n terms, the sum is Sₙ = n(a₁+aₙ)÷2. Flip the sequence over and stack it on top of the original, and the sum of the two numbers at every position always comes out to the same value, a₁+aₙ. The rectangle formed this way has area n×(a₁+aₙ), so the sum of the original sequence is exactly half of that. Gauss's on-the-spot idea became a math formula, exactly as it was.
A geometric sequence grows in a completely different way. Instead of adding the same number every time, it moves forward by multiplying by the same number (the common ratio). Imagine a single bacterium splitting into 2 every minute. 1, 2, 4, 8, 16... it looks slow at first, but after just 20 splits, you're already well past 1 million. If the common ratio is greater than 1, it grows explosively like this; if the common ratio is between 0 and 1 instead (like a radioactive material's half-life, where the amount shrinks by half over time), the value gets closer and closer to 0.
The sum of a geometric sequence has a formula built from a similar idea. It's Sₙ = a₁(rⁿ−1)÷(r−1), and with it you can answer a question like "if you fill each square of a chessboard with grains of rice, doubling each time — 1, 2, 4, 8 — to fill all 64 squares, how many grains in total?" right away, without adding them one by one (for reference, the answer is a staggering number, far more than the entire world's rice production). If the sum of an arithmetic sequence deals with "a world where adding repeats," the sum of a geometric sequence deals with "a world where multiplying repeats."
When studying this with kids, it's most effective to place the two sequences side by side and compare them. For example, draw a table comparing "saving $10 every day" (arithmetic) with "doubling your money every day" (geometric — purely hypothetical, of course!), and you can see with your own eyes that even though the geometric sequence looks slower at first, it overwhelmingly outgrows the other after just a few days. On our activity page, you can change the first term, the common difference (or common ratio), and the number of terms with sliders, and see both the rectangle formed by Gauss's pairing trick and the bar chart of a growing geometric sequence at the same time.