Numerically greatest term in the expansion of $(5+3 x)^6$, when $x=1$. is

Numerically greatest term in the expansion of $(5+3 x)^6$, when $x=1$. is
  1. $3^5 \times 5^3$
  2. $3^3 \times 5^5$
  3. $3^2 \times 5^5$
  4. $3^4 \times 5^4$

Solution

Let $T_{\mathrm{r}+1}$ be the greatest term so, $\frac{T_{r+1}}{T_r} \geq 1$ $\begin{aligned} & \Rightarrow \frac{{ }^6 C_r(5)^{6-r}(3 x)^r}{{ }^6 C_{r-1}(5)^{6-r+1}(3 x)^{r-1}} \geq 1 \\ & \Rightarrow \frac{(r-1)!(7-r)!}{r!\cdot(6-r)!} \times \frac{5^6 \cdot 5^{-r} \cdot 3^r \cdot x^r}{5^7 \cdot 5^{-r} \cdot 3^{r-1} x^{r-1}} \geq 1\end{aligned}$ $\Rightarrow \frac{7-r}{r} \times \frac{3 x}{5} \geq 1 \Rightarrow \frac{3(7-r)}{5 r} \geq 1 \quad(\because$ at,$x=1)$ $\Rightarrow r \leq \frac{21}{8}$ So, $r=2$ Now, $T_{2+1}={ }^6 C_2(5)^{6-2}(3 x)^2=3^3 \times 5^5 . (\because$ at $x=1)$

Asked in: AP EAMCET 2024 (19 May Shift 2)

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