Question

If P(n) : “2.4²ⁿ⁺¹ + 3³ⁿ⁺¹ is divisible by λ for all n ∈ N” is true, then the value of λ is ______.

Answer 1

If P(n): “2.4²ⁿ⁺¹ + 3³ⁿ⁺¹ is divisible by λ for all n ∈ N” is true, then the value of λ is 11.

Explanation:

Now, for n = 1

2.4²⁺¹ + 3³⁺¹ = 2.4³ + 3⁴

= 2.64 + 81

= 128 + 81

= 209

For n = 2,

2.4⁵ + 3⁷ = 8.256 + 2187

= 2048 + 2187

= 4235

Note that the H.C.F. of 209 and 4235 is 11.

So 2.4²ⁿ⁺¹ + 3³ⁿ⁺¹ is divisible by 11.