Foundations — Formulae — Σ1, Σn, Σn², Σn³ — proofs

Before you can read a single line of the parent proofs, you must own every symbol they use. Below, each piece is built from nothing, drawn as a picture, and justified. Read top to bottom — each rung of the ladder rests on the one below it.

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1. A number line — where all our numbers live

Everything we do sits on this road. When we "add up numbers," we are collecting a handful of these marks and asking for their total. Look at the figure: the coloured dots are the numbers $1,2,3,4$ — the very things we will soon be summing.

Why the topic needs it: every symbol below is ultimately a position on this line. If you can point to a number, you can add it.

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2. The counting index $k$ — a placeholder that walks

Why we need it: to describe "add up the first few numbers" without writing them all out, we need a way to say "the number in position $k$." That is the entire job of $k$.

• Other common index letters are $i,j,m$ — same idea, different letter. Choose any letter — meaning is the same.

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3. The letter $n$ — how many terms

Why we need it: without a stopping point, "add up the numbers" never ends. $n$ makes the sum finite and answerable.

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4. The sum sign $\Sigma$ (Sigma) — "add these all up"

This is the star of the whole topic, so we build it slowly.

Look at the figure: the big $\Sigma$ is a machine. You feed in a rule (here, "$k$"), the spotlight walks from the bottom number to the top number, and at each stop it drops a term into a bucket. At the end the bucket's total is the answer.

Why the topic needs it: $\Sigma$ is shorthand. Writing $1+2+\cdots +1000$ is painful; ${\sum }_{k=1}^{1000}k$ says the same thing in one breath — and is the object we will find a shortcut for.

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5. Powers — $k^2$, $k^3$, $k^4$

Why we need it: the four headline sums are $\sum 1$ (power $0$), $\sum k$ (power $1$), $\sum k^2$ (power $2$), $\sum k^3$ (power $3$). Powers are the ingredient list of the whole chapter.

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6. Linearity of $\Sigma$ — pulling constants out and splitting

This is the one rule of manipulation every proof leans on.

Why the topic needs it: every proof mixes $k^2,k,1$ together on one side. Linearity lets us separate them so we can substitute the shorter formulas we already know.

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7. Function notation $f(k)$ — a rule with a name

Why we need it: the master trick (telescoping) is stated as "pick a rule $f$ whose difference is the term you want." We literally cannot say that sentence without the symbol $f(k)$.

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8. The difference $f(k+1)-f(k)$ and telescoping

Look at the figure: the arrows show each $+$ meeting its matching $-$; watch them vanish until only the two endpoints are left standing.

Why the topic needs it: this single collapse is the engine behind all four formula proofs. You choose $f$ cleverly (e.g. $f(k)=k^3$) so that its difference contains the power you want ($k^2$), sum both sides, and the left collapses to just two terms.

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9. Expanding brackets — $(k+1{)}^2$, $(k+1{)}^3$

Why we need it: the differences we telescope, like $(k+1{)}^3-k^3$, must be expanded before the $k^2$ term becomes visible. This connects to Binomial Theorem, the general machine for expanding $(k+1{)}^p$.

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How these feed the topic

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Equipment checklist

Test yourself — cover the right side and answer aloud.

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Connections

• Parent topic → — the four proofs these foundations unlock.
• Telescoping Series — the cancellation engine built in §8.
• Binomial Theorem — the general rule for expanding $(k+1{)}^p$ used in §9.
• Arithmetic Progressions — $\sum k$ is the simplest AP.
• Triangular Numbers — the shape hiding behind $\sum k$.
• Mathematical Induction — a second way to prove each formula.
• Definite Integrals as Limits of Sums — where these sums grow up into areas.