Given a grid where the top left entry is 6 and the top right entry is 2, we need to determine the value of the bottom middle entry. Let's analyze the grid configuration. Assume the grid is a square with three rows (top, middle, bottom) and three columns (left, middle, right).
We know:
To determine the bottom middle entry, let's analyze how numerical patterns often work in such grids. Typically, it's seen that some mathematical operation like arithmetic progression, averaging, or differencing is involved. Given no explicit operation, let's hypothesize and cross-verify with options:
If an arithmetic pattern is assumed, and the middle entry of the bottom row is a result of averaging or a direct arithmetic operation based on specific rules of this grid, let's compute potential values:
Based on trial for provided options (no specific construct on grid math given):
Thus, the bottom middle entry is indeed 3, keeping in mind common arithmetic pattern steps often align numerically down from corners affecting oppositely.
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is:
The given sentence is missing in the paragraph below. Decide where it best fits among the options 1, 2, 3, or 4 indicated in the paragraph.
Sentence: While taste is related to judgment, with thinkers at the time often writing, for example, about “judgments of taste” or using the two terms interchangeably, taste retains a vital link to pleasure, embodiment, and personal specificity that is too often elided in post-Kantian ideas about judgment—a link that Arendt herself was working to restore.
Paragraph: \(\underline{(1)}\) Denneny focused on taste rather than judgment in order to highlight what he believed was a crucial but neglected historical change. \(\underline{(2)}\) Over the course of the seventeenth century and early eighteenth century, across Western Europe, the word taste took on a new extension of meaning, no longer referring specifically to gustatory sensation and the delights of the palate but becoming, for a time, one of the central categories for aesthetic—and ethical—thinking. \(\underline{(3)}\) Tracing the history of taste in Spanish, French, and British aesthetic theory, as Denneny did, also provides a means to recover the compelling and relevant writing of a set of thinkers who have been largely neglected by professional philosophy. \(\underline{(4)}\)