If p(n): 2 4^ 2n-1 + 3^ 3n+1 is divisible by for all n is true, t… - Vidyadip

Question

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

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Answer

$2 \times 42^{n+1}+ 33^{n+1}$
$= 2 \times 24^{n+2} + 3^{3n+1}$
$= 2^{4n+3} + 3^{3n+1}$
Given above expression is divisible by $\lambda$ for all $\text{n}\in\text{N}$
So lets check for $n = 1$
For $n = 2$
$= 2048 + 2187 = 4235$
Now its clearly evident that common factor for above numbers is $11$ so $\lambda = 11$

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