Question

If the sum of the coefficients in the expansion of $\left(1 + 3 x\right)^{n}$ lies between $4000$ and $10000,$ then the value of the greatest coefficient must be

• A. $3954$
• B. $6342$
• C. $4806$
• D. $1458$

Answer 1

Answer: (D)

Putting $x=1$ to get the sum of the coefficients, we get,
$4000 < 4^{n} < 10000\Rightarrow n=6$
$\left(4^{5} = 2^{10} = 1032\right)$
The greatest coefficient is the greatest term in the expansion of $\left(1 + 3 x\right)^{6}$ when $x=1.$
For $\left(1 + 3 x\right)^{6}$ , $m=\frac{\left|3 x\right| \left(6 + 1\right)}{\left|3 x\right| + 1}=\frac{3 \times 7}{4}=5.25$
$\Rightarrow $ Greatest term is $T_{\left[m\right] + 1}=T_{5 + 1}$
$=^{6}C_{5}\left(3 x\right)^{5}=6\times 3^{5}\times x^{5}$
Greatest coefficient $=6\times 3^{5}=1458$