Using binomial expansion of ${{\left( 1+x \right)}^{n}}$, or otherwise, show that \({{7}^{n}}\left

Question

Using binomial expansion of ${{\left( 1+x \right)}^{n}}$, or otherwise, show that ${{7}^{n}}\left( 3n+1 \right)-1$ is always divisible by $9$, for integer $n\ge 1$, ${{9}^{n}}-8n-1$ is divisible by $64$, for integer $n\ge 1$.

Accepted Answer

(i) $9\mid\left(7^n(3n+1)-1\right)$ for every integer $n\ge 1$ (ii) $64\mid(9^n-8n-1)$ for every integer $n\ge 1$

Binomial Theorem - Divisibility Proofs Using Binomial Expansions

Binomial Theorem - Divisibility Proofs Using Binomial Expansions

Video Solution:

Solution:

Solution locked