← All parts of this equation

Equation 11 · Part 13 · The Unit Economics of a Token

Numerator: k (k+1)

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

What this part means

The complete quantity above the fraction bar.

Its job in the formula

k (k+1) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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 Tin(k)=∑j=1k(b+ja)=kb+a k(k+1)2T_{\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 input billing. This is why cache hit rate dominates agentic economics: prefix caching converts most of that quadratic term into reads at one tenth the price, and a workload that loses its cache to a time-to-live expiry or an effort change pays the full quadratic […

Read this part in the article →

Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.