Finding the general formula for $a_{n+1}=2^n a_n +4$, where $a_1=1$.

Question

Problem:

1. Find the general formula for $a_{n+1}=2^n a_n +4$, where $a_1=1$.
2. Find the sum of its first $2n$ terms with odd subscript.

My effort:

• It seems to me that $a_{n+1} / 2^{(n+1)^2/2}=\dfrac{1}{\sqrt{2}}a_n/2^{n^2/2} +4/ 2^{(n+1)^2/2}$, which is $b_{n+1}=\dfrac{1}{2^{1/2}} b_{n} + \dfrac{1}{2^{(n+3)(n-1)/2}}$, where $b_n=a_n/2^{n^2/2}$. But it seems hard to deal with the last term.
• The first ten $a_n$ is {1, 6, 28, 228, 3652, 116868, 7479556, 957383172, 245090092036, 125486127122436}, which follows no immediate rule.
• Write the sequence in binary form, I find it {1, 110, 11100, 11100100, 111001000100, ...} which is generally in a 1 2*0 1 3*0 1 4*n ... pattern (apart from the first few). So I highly suspect that there is not closed form expression. But how to prove this?

Answer 1

Here is a solution by substitution: \begin{align} a_1&=1\\ a_2&=2^1+4\\ a_3&=2^{1+2}+4(1+2^2)\\ a_{1+3}&=2^{1+2+3}+4(1+2^3+2^{3+2})\\ a_{1+4}&=2^{1+2+3+4}+4(1+2^4+2^{4+3}+2^{4+3+2})\\ a_{1+5}&=2^{1+2+3+4+5}+4(1+2^5+2^{5+4}+2^{5+4+3}+2^{5+4+3+2})\\ a_{1+6}&=2^{1+2+3+4+5+6}+4(1+2^6+2^{6+5}+2^{6+5+4}+2^{6+5+4+3}+2^{6+5+4+3+2})\\ ...\\ a_{1+n}&=2^{\sum_{j=1}^nj}+4(1+\sum_{j=1}^{n-1}2^{\sum_{k=j+1}^nk}) \end{align} I should be simplifying this last expression ... but I could not ... :-)

Answer 2

\begin{align*} a_{n+1}&=2^{n}a_{n}+4\\&=2^{n+(n-1)}a_{n-1}+4\cdot 2^{n}+4\\&=2^{n+(n-1)+(n-2)}a_{n-2}+ 4\cdot 2^{n+(n-1)} + 4\cdot 2^n + 4\\&=2^{n(n+1)/2}+4(1+\sum_{j=0}^{n-1} 2^{\sum_{k=j+1}^n k})\\&=2^{n(n+1)/2}+4(1+\sum_{j=0}^{n-1} 2^{n(n+1)/2-j(j+1)/2})\\&=2^{n(n+1)/2}+4(1+2^{n(n+1)/2}\sum_{j=0}^{n-1} 2^{-j(j+1)/2}) \end{align*}

I'm not sure how to evaluate this last sum. Maybe someone else can assist?

Answer 3

You can proceed backward in order to "guess" the solution and then prove it properly by induction. I mean: $a_n=2^{n-1}a_{n-1}+4=2^{n-1}(2^{n-2}a_{n-2}+4) + 4=2^{(n-1)+(n-2)}a_{n-2}+4*2^{n-1}+4=\dots$

Doing this you can see what it will look like when you have $a_1$ instead of $a_{n-2}$ (you can make a few more steps if necessary). Once it's done, you just have to simplify it, then prove it by induction if you want to be really rigorous.