Related series and sequences

In case 2 ignore the first term 1 , but take the first term (1) of case 1 in account, and consider 2 as first term 4 as second term etc..,.

If $a_n$ and $b_n$ are $n^{th}$ terms of case 1 and case 2 respectively, and $A_n$ and $B_n$ represents sum to $n^{th}$ term of case 1 and case 2 respectively, find the value of $B_{n-1} - A_n$