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Published equation contexts

B(k)=Δ∑i=1ki=Δ k(k+1)2B(k) = \Delta \sum_{i=1}^{k} i = \Delta\,\frac{k(k+1)}{2}

Why this formula appears here

Model it simply. Suppose a session runs for k turns, and each turn adds on average Δ\Delta tokens of new tool result to a transcript that is resent in full on every request, because nothing is evicting or compressing it. The tokens billed on turn i include roughly iΔ\Delta tokens of accumulated result history, so the total billed across the whole session is B(k)=Δ∑i=1ki=Δ k(k+1)2B(k) = \Delta \sum_{i=1}^{k} i = \Delta\,\frac{k(k+1)}{2}. Doubling the length of a session does not double the bill; it roughly quadruples it, because the earlier results are being paid for again on every one of the later turns. This is not a hypothetical — it is the direct arithmetic consequence of the same design choice OpenAI documents for its own tool schemas: whatever is…

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ii

Symbol i

i appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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i=1i=1

Starting index or lower bound: i=1

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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kk

Ending index or upper bound: k

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

B(k)=Δ∑i=1ki=Δ k(k+1)2.B(k) = \Delta \sum_{i=1}^{k} i = \Delta\,\frac{k(k+1)}{2}.

Equation 10 · AI Infrastructure

The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Model it simply. Suppose a session runs for k turns, and each turn adds on average Δ\Delta tokens of new tool result to a transcript that is resent in full on every request, because nothing is evicting or compressing it. The tokens billed on turn i include roughly iΔ\Delta tokens of accumulated result history, so the total billed across the whole session is B(k)=Δ∑i=1ki=Δ k(k+1)2B(k) = \Delta \sum_{i=1}^{k} i = \Delta\,\frac{k(k+1)}{2}. Doubling the length of a session does not double the bill; it roughly quadruples it, because the earlier results are being paid for again on every one of the later turns. This is not a hypothetical — it is the direct arithmetic consequence of the same design choice OpenAI documents for its own tool schemas: whatever is…

Meanings in this article

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