Question

If P(n): 2 × 42^{n + 1} + 3^{3n + 1} is divisible by λ for all n ∈ N is true, then find the value of λ.

Answer 1

for n=1, 

2×4^2×1+1 + 3^3×1+1=2×4³+3⁴ 

= 2×64+81 

= 128+81 

= 209 

For n=2, 

2×4^2×2+1 + 3^3×2+1 = 2×4⁵+3⁷ 

= 2×1024+2187 

= 2048+2187 

= 4235 

Now, the H.C.F of 209 and 4235 is 11. 

Hence, λ=11.