Answer:
Proved (See Explanation)
Step-by-step explanation:
Show that 3ⁿ⁺⁴ - 3ⁿ is divisible by 16.
This is done as follows
[tex]\frac{3^{n+4} - 3^n}{16}[/tex]
From laws of indices;
aᵐ⁺ⁿ = aᵐ * aⁿ.
So, 3ⁿ⁺⁴ can be written as 3ⁿ * 3⁴.
[tex]\frac{3^{n+4} - 3^n}{16}[/tex] becomes
[tex]\frac{3^n * 3^4 - 3^n}{16}[/tex]
Factorize
[tex]\frac{3^n(3^4 - 1)}{16}[/tex]
[tex]\frac{3^n(81 - 1)}{16}[/tex]
[tex]\frac{3^n(80)}{16}[/tex]
3ⁿ * 5
5(3ⁿ)
The expression can not be further simplified.
However, we can conclude that when 3ⁿ⁺⁴ - 3ⁿ is divisible by 16, because 5(3ⁿ) is a natural whole number as long as n is a natural whole number.
