Question

If the sum of the coefficients in the expansion of $(1+2x{)}^n$ is $2,187$, the greatest term in the expansion, if $x=\frac{1}{2}$ is/are

• A. $5^{th}$
• B. $4^{th}$ and $5^{th}$
• C. $4^{th}$
• D. None of these

Answer 1

The correct option is B $4^{th}$ and $5^{th}$

Substituting $x=1$ for the sum of coefficients of the given expansion.
$3^n=2187$
${\log }_3\left(n\right)={\log }_3\left(2187\right)$
$n=7$
Hence the expansion will be of the form $(1+2x{)}^7$
Now there will be total of 8 terms, and for $x=\frac{1}{2}$ the value of a term will only depend on its binomial coefficient.
Therefore the greatest terms will be the set of two middle terms. which are the 4th term and the 5th term.