Question

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

Answer 1

$=2\times 4^{2n+1}+3^{3n+1}$

$=2\times 4^{4n+2}+3^{3n+1}$

$=4^{4n+3}+3^{3n+1}$

Given above expression is divisible by $\lambda$ for all $nϵN$

so lets check for n = 1

$=128+81=209$

for n = 2

$=2048+2187=4235$

Now its clearly evident that common factor for above numbers is 11 so $\lambda =11$.