Equation 11 · Part 12 · The Unit Economics of a Token

Ending index or upper bound: k

T_{\mathrm{in}}(k) = \sum_{j=1}^{k} \left( b + j a \right) = k b + a \, \frac{k (k+1)}{2}

k

What this part means

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

Its job in the formula

k occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Full expressionEnding index or upper bound: kArticle meaning

The passage around this formula

The nonlinearity that surprises people does not live in this equation, which is linear. It lives in agent loops. Suppose a loop runs for k steps, each step appends roughly a tokens of tool output and assistant text, and the context started at b tokens. The total input billed across the whole loop is then $T_{\mathrm{in}}(k) = \sum_{j=1}^{k} \left( b + j a \right) = k b + a \, \frac{k (k+1)}{2}$ . which is quadratic in the number of steps. Doubling the length of an agent trajectory roughly quadruples its…

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Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the article section

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