Question

The greatest term in the expansion of ${(1+3\mathrm{x})}^{54}$, where $x=\frac{1}{3}$ is

• A. ${\mathrm{T}}_{28}$
• B. ${\mathrm{T}}_{25}$
• C. ${\mathrm{T}}_{26}$
• D. ${\mathrm{T}}_{24}$

Answer 1

The correct option is A

${\mathrm{T}}_{28}$

The explanation for the correct option

Find the greatest term in the expansion.

We know $\frac{\mathrm{n}-\mathrm{r}+1}{\mathrm{r}}\ge 1$ is the greatest term in the expansion of ${(1+3\mathrm{x})}^{54}$.

$$\begin{gathered} \Rightarrow \frac{54-\mathrm{r}+1}{\mathrm{r}}\ge 1 \\ \Rightarrow 55-\mathrm{r}\ge \mathrm{r} \\ \Rightarrow 55\ge 2\mathrm{r} \\ \Rightarrow \mathrm{r}=27 \end{gathered}$$

Therefore $28\mathrm{th}$term is the greatest term.

Hence option $A$ is the correct option.