Arithmetic and geometric sequences

Parametric equations

An arithmetic sequence $a_n=a_1+(n-1)d$ is a row of bars whose heights lie on one slanted line. A geometric sequence $a_n=a_1{r}^{n-1}$ is a row of rectangles of equal height, with each width scaled to that term. The number beside the picture is the closed form of $S_n$.

Arithmetic sum:

Geometric sum ($r\ne 1$):

Interaction

• Arithmetic / Geometric: switch the sequence. The sliders are then First term a₁, Common difference d, and Number of terms n, or First term a₁, Common ratio r, and Number of terms n
• First term a₁: move the start, and watch the whole sequence and $S_n$ change with it
• Common difference d / Common ratio r: change how the terms grow, and compare the linear stack with widths that shrink by the ratio
• Number of terms n: change $n$, and compare the picture with the formula value of $S_n$

What to notice

• In the arithmetic view, the bar heights lie on one slanted line, and an outline of the same terms in reverse sits on that same row.
• In the geometric view, every rectangle has the same height and a width scaled to that term. As the terms appear, the total area adds up to the closed form $S_n$.
• When $n$ increases, the picture builds the sum term by term, and the formula value is the one shown with the controls.

Related

• Fibonacci spiral
• Basel problem
• Geometric significance of the binomial expansion

Further reading

• Arithmetic progression (Wikipedia)
• Geometric progression (Wikipedia)