Adding every number from 1 to 100 one by one is quite a task. With arithmetic and geometric sequences, you can find the sum directly with a single formula, no matter how many terms there are. Adjust the first term, common difference (or common ratio), and number of terms with the sliders, and check that the stacked bars always match the value calculated by the formula.
Why the arithmetic-sequence sum formula works (Gauss’s method): When finding the sum from 1 to 100, pair the numbers from the two ends: 1+100, 2+99, 3+98, and so on. Every pair is 101. There are 50 such pairs, so the sum is 101×50. Generalizing this gives S_n = n(a₁+aₙ)/2.
Why the geometric-sequence sum formula works: Multiply S_n by the common ratio r to get rS_n, then subtract it from S_n. Most terms cancel, leaving only the parts involving the first and last terms. This gives S_n = a₁(rⁿ−1)/(r−1), where r≠1.