Answer:
Proved (See Explanation)
Explanation:
Show that 3ⁿ⁺⁴ - 3ⁿ is divisible by 16.
This is done as follows
![(3^(n+4) - 3^n)/(16)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/2gjmst0rbjthgclqm1budprn69i0ny1qye.png)
From laws of indices;
aᵐ⁺ⁿ = aᵐ * aⁿ.
So, 3ⁿ⁺⁴ can be written as 3ⁿ * 3⁴.
becomes
![(3^n * 3^4 - 3^n)/(16)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/48dvn337i8cb79jwhmilzlq7hter5yxygi.png)
Factorize
![(3^n(3^4 - 1))/(16)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/hcjblzudusgx7q2elobasn9scd1bkbj91b.png)
![(3^n(81 - 1))/(16)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/an1r922qlnyslnv674lfupp7j71akol74b.png)
![(3^n(80))/(16)](https://img.qammunity.org/2021/formulas/mathematics/middle-school/r709v5ccbrewxe8exzli5uk0r2a4apgal2.png)
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.