Equation 4 · Actually Building with the Grok API
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol p_o^L
is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
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What the article says around this equation
The pricing structure has a feature worth building a mental model around before it costs you money in production: it is not a marginal rate that blends smoothly as a request grows, it is a cliff. Every model’s documented price list carries two tiers — one for requests under 200,000 tokens, and a materially higher one, roughly double across the board, once a request reaches that threshold — and the higher rate applies to the entire request, not just the tokens past the line [ 1 ] . Written as a cost function of input tokens and output tokens against a threshold T = 200{,}000 , with low-tier prices , and high-tier prices , :
Sources cited in the surrounding passage
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