Prove , by mathematical induction , that 8.7ⁿ+4ⁿ⁺² is divisible by 24 but not 48 for all n

Let the given statement be P(n) i.e.,

P(n) = 8.7ⁿ+4ⁿ⁺² 

for n= 1, 

P(1) = 8.7+4³ = 56+64 = 120, which is divisible by 24 not by 48.

Let P(k) be true for some positive integer k, i.e.

P(k) = 8.7^k+4^k+2 

⇒8.7^k+4^k+2  = 24λ

now, prove that P(k + 1) is true.

, which is divisible by 24 not by 48.

therefore , the result is true n=k+1.

therefore, if the result is true for n = k, then it is also true for n = k+1.

but the result is true for n = 1.

hence, by PMI , result is true for all n