<s>[INST] Divisibility by 7 Proof by Induction Prove by Induction that
$$7|4^{2^{n}}+2^{2^{n}}+1,\; \forall n\in \mathbb{N}$$
Base case:
$$
\begin{aligned}
7&|4^{2^{1}}+2^{2^{1}}+1,\\
7&|7\cdot 3
\end{aligned}$$ Which is true.
Now, having $n=k$, we assume that:
$$7|4^{2^{k}}+2^{2^{k}}+1,\;\; \forall k\in \mathbb{N}$$
We have to prove that for $n=k+1$ that,
$$7|4^{2^{k+1}}+2^{2^{k+1}}+1,\;\; \forall k\in \mathbb{N}$$
We know that if $a$ is divisible by 7 then $b$ is divisible by 7 iff $b-a$ is divisible by 7.
Then,
$$
\begin{aligned}
b-a &= 4^{2^{k+1}}+2^{2^{k+1}}+1 - (4^{2^{k}}+2^{2^{k}}+1)\\
&= 4^{2^{k+1}}+2^{2^{k+1}} - 4^{2^{k}}-2^{2^{k}}\\
&= 4^{2\cdot  2^{k}}+2^{2\cdot  2^{k}} - 4^{2^{k}}-2^{2^{k}}
\end{aligned}
$$
I get stuck here, please help me.
 [/INST] Other answers have shown how you could continue with what you had. But just for interest, here's another proof, which doesn't use the difference, but the quotient:
Let's call the formula
$$ a_n = 4^{2^n} + 2^{2^n} + 1 $$
You already showed that $a_1 = 21$ is a multiple of $7$. (So is $a_0 = 4^1+2^1+1 = 7$, so we can actually prove it for every non-negative integer $n$.)
With so many $2$'s involved, a substitution $4=2^2$ seems worth trying:
$$ a_n = (2^2)^{2^n} + 2^{2^n} + 1 = 2^{2 \cdot 2^n} + 2^{2^n} + 1 $$
The multiple exponent operations are a bit confusing, but we can abstract them out by noticing $2^{2 \cdot 2^n} = (2^{2^n})^2$, so if we define $x_n = 2^{2^n}$, then
$$ a_n = x_n^2 + x_n + 1 $$
Take $7 | a_n$ as the inductive hypothesis, and look at the next iteration:
$$ x_{n+1} = 2^{2^{n+1}} = 2^{2 \cdot 2^n} = (2^{2^n})^2 = x_n^2 $$
$$ a_{n+1} = x_{n+1}^2 + x_{n+1} + 1 = x_n^4 + x_n^2 + 1 $$
But this factors as
$$ a_{n+1} = (x_n^2 - x_n + 1) (x_n^2 + x_n + 1) = (x_n^2 - x_n + 1) a_n $$
Since $x_n$ is a positive integer, $(x_n^2 - x_n + 1)$ is also a positive integer. Since $7 | a_n$, we also have $7 | a_{n+1}$. Therefore by induction, $7 | a_n$ for every natural $n$.
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