Question

If $P\left(n\right):{2.4}^{2n+1}+3^{3n+1}$ is divisible by $\lambda$ for all $nϵN$ is true then the value of $\lambda$ is ______.

Answer 1

${2.4}^{2n+1}+3^{3n+1}\Rightarrow {\mathrm{2.4.4}}^{2n}+{3.3}^{3n}\Rightarrow {8.4}^{2n}+{3.3}^{3n}$

Now to check the minimum value of $\lambda$

Put $n=0$

$\Rightarrow 8+3=11$

Hence, value of $\lambda =11$