Question

If P(n) : "2 × 4^{2n + 1} + 3^{3n + 1} is divisible by λ for all n ∈ N" is true, then the value of λ is _____________.

Answer 1

If P(n) = 2 × 4^{2n + 1} + 3^{3n + 1} is divisible by λ ∀ n ∈ N is true

for n = 1,
$$\begin{gathered} P\left(1\right):2\times 4^{2\left(1\right)+1}+3^{3\left(1\right)+1} \\ =2\times 4^3+3^4 \\ =2\times 64+81 \\ =128+81 \\ P\left(1\right)=209 \end{gathered}$$

for n = 2,
$$\begin{gathered} P\left(2\right):2^3\times 4^{2\left(2\right)+1}+3^{3\left(2\right)+1} \\ =2^3\times 4^5+3^7 \\ =2^3\times 256+2187 \\ =2048+2187 \\ P\left(2\right)=4235 \end{gathered}$$
Since common factor of P(1) and P(2) is 11
Hence 2 × 4²ⁿ⁺¹ + 3³ⁿ⁺¹ is divisible by 11
i.e λ = 11