Question

If the sum of the coefficients in the expansion of ${\left(1+3x\right)}^n$ 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 ${\left(1+3x\right)}^6$ when $x=1.$
For ${\left(1+3x\right)}^6$ , $m=\frac{|3x|\left(6+1\right)}{|3x|+1}=\frac{3\times 7}{4}=5.25$
$\Rightarrow$ Greatest term is $T_{\left[m\right]+1}=T_{5+1}$
${=}^6{C}_5{\left(3x\right)}^5=6\times 3^5\times x^5$
Greatest coefficient $=6\times 3^5=1458$