If P(n): 2 × 42n + 1 + 33n + 1 is divisible by λ for all n ∈ N is true, then find the value of λ.
for n=1,
2×42×1+1 + 33×1+1=2×43+34
= 2×64+81
= 128+81
= 209
For n=2,
2×42×2+1 + 33×2+1 = 2×45+37
= 2×1024+2187
= 2048+2187
= 4235
Now, the H.C.F of 209 and 4235 is 11.
Hence, λ=11.