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**Soltras** Hi Lisa.

Use your inductive assumption.

You want to see if the statement is true for $\displaystyle n=k+1$.

Well you seem to have correctly observed that the expression for $\displaystyle n = k+1$ is just the expression for $\displaystyle n = k$ with $\displaystyle (k+4)^4-k^4$ added on.

Since by inductive hypothesis, 16 divides the expression for $\displaystyle n = k$, all you have to do is verify that 16 divides $\displaystyle (k+4)^4-k^4$.

Expand this and you will see every term is divisible by 16. (you may find it easy to expand if you take advantage of its form as a difference of squares).

In your proof, you might want to explain that if a number (16) divides two numbers, then it divides their sum as well. Presumably, this property has been discussed/proved in your class or textbook by now, so just mentioning the fact will probably suffice.