Greatest Term

The greatest term in a numerical binomial expansion is located by examining the ratio of a term to its predecessor and finding where this ratio stops exceeding one, a classic JEE Advanced optimisation idea. For (1 + x)^n the ratio is T_{r+1}/T_r = [C(n, r)/C(n, r-1)] x = [(n - r + 1)/r] x. The terms increase as long as this ratio is at least one, that is while T_{r+1}/T_r >= 1, and they start decreasing once the ratio drops below one; the greatest term occurs at the last r for which the ratio is still greater than or equal to one. Hence the governing condition is T_{r+1}/T_r >= 1. Option T_{r+1}/T_r = n has no general validity and confuses the ratio with the exponent. Option T_{r+1} - T_r = 0 always is false because terms genuinely change in size. Option T_{r+1}/T_r <= 0 is impossible for positive terms. The transition of the ratio through one marks the peak. Plausibility check: since the ratio decreases steadily in r, there is a unique crossover, guaranteeing a well-defined greatest term, consistent with the unimodal nature of the expansion.