Question

Using binomial theorem, prove that (2³ⁿ - 7n -1) is divisible by 49, where n N.

Answer 1

To prove: (2³ⁿ - 7n -1) is divisible by 49, where n N 

Formula used: (a+b)ⁿ = ⁿC₀aⁿ + ⁿC₁a ^n-1b + ⁿC₂a ^n-2b² + …… +ⁿC^n-1ab^n-1 + ⁿC_nbⁿ

(2³ⁿ - 7n -1) = (2³)ⁿ – 7n – 1 

⇒ 8ⁿ - 7n – 1 

⇒ (1+7)ⁿ – 7n – 1

Now, (2³ⁿ - 7n -1) = 49K 

Therefore (2³ⁿ - 7n -1) is divisible by 49