How do you generalised the formula of expanded terms $\frac{((50^A)*(((\frac{(50^B)((\frac{(50^C)+1)}{(4)}))+1)}{(4)}))}{(4)}$

Question

I don't know where to start with this, is it proof by induction? I am trying to create a generalised formula for the expanded brackets and prove that it works for any number of nested terms like the above would be

.

If I increased the terms for example I make the powers of 50 go ABCDEFG and so on (where A starts from the out side and works inwards) following the pattern of +1 and divide by 4. What is the general formula and how can I prove it? Can you walk me through the steps.

Answer 1

Let $\{A_i\}_{i \in \mathbb{N}}$ your sequence of $A$, $B$ , $C$ ...

Let $$ f_n : x\to \dfrac{x^{A_n}+1}{4}$$

$$ g_n: (x,y) \to \dfrac{x^{A_n}y+1}{4} $$

Given the $n \in \mathbb{N}$

Let's call $P_n(x)$ the n-th iteration of your process.

$$ P_1(x)=\dfrac{x^{A_1}+1}{4}=g_1(x,1)$$

$$ P_2(x)=\dfrac{x^{A_1}\left(\dfrac{x^{A_2}+1}{4}\right)+1}{4}=g_1(x,g_2(x,1))$$

$$ P_3(x) = \dfrac{x^{A_1}\dfrac{x^{A_2}\left(\dfrac{x^{A_3}+1}{4}\right)+1}{4}+1}{4}=g_1(x,g_2(x,g_3(x,1))) $$

So

$$ P_n(x)=g_1(x,g_2(x,g_3(x,g_4(..g_n(x,1)))...)$$