ALG-11

Geometric series

S = a (1 − r^n) / (1 − r) for r ≠ 1. n terms, first term a.

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SeriesGeometric series

Governing equation

S=a\dfrac{1-r^n}{1-r}

$n \in \left[1, 40\right]$

where

a: First term (—)
r: Ratio (—)
n: Terms (—)
S: Sum (—)

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\dfrac{1-r^n}{1-r}

$n \in \left[1, 40\right]$

$n = 1$ $S = 2$

Along the curve

②

Domain of the variable — blue

n_{\min} = 1,\quad n_{\max} = 40

$n \in \left[1, 40\right]$

$n = 1$ $n = 1$

Along the curve

③

Response curve — green

S=a\dfrac{1-r^n}{1-r}

$n \in \left[1, 40\right]$

$n = 1$ $S = 2$

Along the curve

④

Rate of change — amber

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

$n \in \left[1, 40\right]$

$n = 1.245$ $S^{\prime} = 0$

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-11 — Geometric series) is the form associated with Geometric series. Working symbols: $a$ , $r$ , $n$ $\rightarrow$ $S$ . Multiply by r and subtract: the interior terms cancel. |r| < 1 gives the infinite sum a/(1−r).

Purpose

Purpose: compute

S

from

a

r

n

in Algebra via

S=a\dfrac{1-r^n}{1-r}

S = a (1 − r^n) / (1 − r) for r ≠ 1. n terms, first term a. 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 = 2.000\,\mathrm{—}

r = 0.500\,\mathrm{—}

n = 8.000\,\mathrm{—}

, the governing relation

S=a\dfrac{1-r^n}{1-r}

yields

S = 3.9844\,\mathrm{—}

. Finite n, r ≠ 1 handled; r ≈ 1 falls back to n a. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

First term a2 —Ratio r0.5 —Terms n8 —

Outputs

Sum S3.9844 —

Reading speed

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Narration of this film

Finite n, r ≠ 1 handled; r ≈ 1 falls back to n a.

Multiply by r and subtract: the interior terms cancel. |r| < 1 gives the infinite sum a/(1−r).

Reading speed

Watch on YouTube