With binomial theorem when it ask for greatest term, is it asking for coefficient? Because I always get the number that have the highest coefficient when solving with but the book is telling me the answer being the term higher than that
e.g. from old fitz
Once you sub in \( x = \frac12\) things change. The greatest coefficient is used in the case where \(x\) is unknown, and we just care about the coefficients. The greatest term, however, becomes a thing once we actually sub \(x\) in for something.
For that question, if we just wanted the greatest coefficient, we'd take \(T_k \) to just be the coefficient, and nothing else. i.e. \(T_{k+1}= \binom{12}{k} 3^{12-k} 4^k \).
But if we wanted the greatest term after subbing in \( x = \frac12\), we'd have to consider the actual term \(T_{k+1} = \binom{12}{k}3^{12-k}4^k x^k\), but after subbing in \(x = \frac12\). That is, we'd be considering \(T_{k+1} = \binom{12}{k} 3^{12-k} 4^k \left(\frac12 \right)^k \)