The middle term in the expansion of ${(1 + x)^{2n}}$ is

The value of $x$, for which the 6th term in the expansion of ${\left\{ {{2^{{{\log }_2}\sqrt {({9^{x – 1}} + 7)} }} + \frac{1}{{{2^{(1/5){{\log }_2}({3^{x – 1}} + 1)}}}}} \right\}^7}$ is $84$, is equal to

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2 x)^{ n }$ be in the ratio $2: 5: 8$. Then the coefficient of the term, which is in the middle of these three terms, is $………..$.

Find the $4^{\text {th }}$ term in the expansion of $(x-2 y)^{12}$

If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^{n}-b^{n}$, whenever $n$ is a positive integer.