Answer :
Answer:
Step-by-step explanation:
Prove that for any natural number 3^(n+4)-3n is divisible by 16.
(I'm going to assume that you mean 3^(n+4)-3^n.)
1. We can break up 3^(n+4)-3n into 3^n * 3^4-3^n (by the rule a^b*a^c = a^b+c).
2. Solve to get 3^n * 81 - 3^n
3. Factor out the 3^n, and you'll get 3^n(81-1), and simplify: 3^n(80)
You may notice that 80 is divisible by 16.
4. Rewrite what we got from the last step as: 3^n*5(16).
Hope this helped you!
The proof that any natural number n in (3ⁿ⁺⁴ - 3ⁿ) is divisible by 16 is; it was simplified to 3ⁿ × 5
What is the proof that the number is divisible?
We want to show that show that 3ⁿ⁺⁴ - 3ⁿ is divisible by 16.
This is; (3ⁿ⁺⁴ - 3ⁿ)/16
From laws of indices, we know that;
aᵐ⁺ⁿ = aᵐ × aⁿ.
Thus; 3ⁿ⁺⁴ can be expressed as 3ⁿ × 3⁴.
Thus; (3ⁿ⁺⁴ - 3ⁿ)/16 is;
((3ⁿ × 3⁴) - 3ⁿ) = 3ⁿ((81 - 1)/16)
3ⁿ(80/16) = 3ⁿ × 5
The expression has been fully simplified to show that any natural number n in (3ⁿ⁺⁴ - 3ⁿ) is divisible by 16.
Read more about proof of numbers at; https://brainly.com/question/10657323