ALG-23

Geometric sum

Finite geometric.

Reading speed

Algebrageometric series

Governing equation

S=a(1-r^n)/(1-r)

$n \in \left[1, 30\right]$

where

a: a (—)
r: r (—)
n: n (—)
S: S (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

S=a(1-r^n)/(1-r)

$n \in \left[1, 30\right]$

$n = 1$ $S = 1$

Along the curve

②

Domain of the variable — blue

n_{\min} = 1,\quad n_{\max} = 30

$n \in \left[1, 30\right]$

$n = 1$ $n = 1$

Along the curve

③

Response curve — green

S=a(1-r^n)/(1-r)

$n \in \left[1, 30\right]$

$n = 1$ $S = 1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}S}{\mathrm{d}n}

$n \in \left[1, 30\right]$

$n = 1.182$ $S^{\prime} = 0.612$

Along the curve

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-23 — Geometric sum) is the form associated with geometric series. Working symbols: $a$ , $r$ , $n$ $\rightarrow$ $S$ . Finite geometric.

Purpose

Purpose: compute

S

from

a

r

n

in Algebra via

S=a(1-r^n)/(1-r)

Finite geometric. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 1.000\,\mathrm{—}

r = 0.500\,\mathrm{—}

n = 8.000\,\mathrm{—}

, the governing relation

S=a(1-r^n)/(1-r)

yields

S = 1.9922\,\mathrm{—}

. Finite geometric. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a1 —r r0.5 —n n8 —

Outputs

S S1.9922 —

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Narration of this film

Finite geometric.

Reading speed

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