ALG-12

Arithmetic series

S = n/2 (2a + (n−1)d). n terms, first a, common difference d.

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SeriesArithmetic series

Governing equation

S=\dfrac n2\bigl(2a+(n-1)d\bigr)

$n \in \left[1, 80\right]$

where

a: First term (—)
d: Difference (—)
n: Terms (—)
S: Sum (—)
\ell: Last term (—)

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=\dfrac n2\bigl(2a+(n-1)d\bigr)

$n \in \left[1, 80\right]$

$n = 1$ $S = 3$

Along the curve

②

Domain of the variable — blue

n_{\min} = 1,\quad n_{\max} = 80

$n \in \left[1, 80\right]$

$n = 1$ $n = 1$

Along the curve

③

Response curve — green

S=\dfrac n2\bigl(2a+(n-1)d\bigr)

$n \in \left[1, 80\right]$

$n = 1$ $S = 3$

Along the curve

④

Rate of change — amber

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

$n \in \left[1, 80\right]$

$n = 1.497$ $S^{\prime} = 5.032$

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-12 — Arithmetic series) is the form associated with Arithmetic series. Working symbols: $a$ , $d$ , $n$ $\rightarrow$ $S$ , $\ell$ . Gauss pairing: first + last, n/2 times. Also S = n (a + ℓ)/2.

Purpose

Purpose: compute

S

\ell

from

a

d

n

in Algebra via

S=\dfrac n2\bigl(2a+(n-1)d\bigr)

S = n/2 (2a + (n−1)d). n terms, first a, common difference d. 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 = 3.000\,\mathrm{—}

d = 2.000\,\mathrm{—}

n = 10.000\,\mathrm{—}

, the governing relation

S=\dfrac n2\bigl(2a+(n-1)d\bigr)

yields

S = 120.000\,\mathrm{—}

\ell = 21.000\,\mathrm{—}

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

Calculator

Inputs

First term a3 —Difference d2 —Terms n10 —

Outputs

Sum S120.000 —
Last term \ell21.000 —

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Narration of this film

Finite arithmetic progression.

Gauss pairing: first + last, n/2 times. Also S = n (a + ℓ)/2.

Reading speed

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