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Integration by Substitution

Find the nume...

Question

Find the numerically greatest term in the expansion of $(2 x + y)^{14}$ when $x = 4$ and $y = 5$ .

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Solution

$\begin{matrix} T_{r + 1} \geq T_{n}^{n} C_{r} {(2 x)}^{r} y^{n - r} \geq^{n} C_{r - 1} {(2 x)}^{r - 1} y^{n - r + 1} \frac{^{n} C_{r}}{^{n} C_{r - 1}} \geq \frac{y}{2 x} \frac{^{14} C_{r}}{^{14} C_{r - 1}} \geq \frac{5}{8} \frac{\frac{14!}{r! (14 - r)!}}{\frac{14!}{(r - 1)! (14 - r + 1)!}} \geq \frac{5}{8} \frac{14 - r + 1}{r - 1} \geq \frac{5}{8} 8 (14 - r + 1) \geq 5 (r - 1) 112 - 8 r + 8 \geq 5 r - 1 13 r \leq 125 r \leq \frac{125}{13} r_{max} = 9 \end{matrix}$
So, the greatest term is $T_{10}$

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