For Set of number card $$A=\{1,2,3,4\}$$ we pick one card in $A$ at a time. When we do this $n$ time, $a_n$ is Number of case that sum of all number cards's that we picked is multiples of 5. Find $$\sum\limits_{n = 1}^5 \sqrt{a_n +a_{n+1}}$$
- My way is $a_n +a_{n+1}=4^n$ because of one by one bijection. So it is $\sum\limits_{n = 1}^5 \sqrt{a_n +a_{n+1}} =2+2^2+2^3+2^4=62$
But I couldn't get $n$ th term of $a_n.$
Hint: Using your formula, and starting with $a_1=0$, we get the sequence $$0,4,16-4,64-16+4,256-64+16-4,...$$ Do you know geometric series ?