Question

If P(n): $2\times 4^{2n+1}+3^{3n+1}$ is divisible by $\lambda$ for all n $\in$ N is true, then find the value of $\lambda$.

Answer 1

$$\begin{gathered} \mathrm{For}n=1, \\ \mathrm{P}\left(1\right)=2\times 4^{2+1}+3^{3+1}=2\times 4^3+3^4=128+81=209 \\ \mathrm{For}n=2, \\ \mathrm{P}\left(2\right)=2\times 4^{4+1}+3^{6+1}=2\times 4^5+3^7=2048+2187=4235 \\ \mathrm{As},\mathrm{HCF}\left(209,4235\right)=11 \\ \mathrm{So},2\times 4^{2n+1}+3^{3n+1}\mathrm{is}\mathrm{divisible}\mathrm{by}11. \\ \mathrm{Hence},\mathrm{the}\mathrm{value}\mathrm{of}\lambda \mathrm{is}11. \end{gathered}$$