What's next?

Using: a^x-b^x we have= 1.-a^0-b^0=1-1=0 2.-a^1-b^1=a-b=1 3.-a^2-b^2= 7 diviging the third equation by the first one we have: a^2-b^2 / a-b = 7/1 4.-a+b =7 using the forth and the second equation we find out that a=4 and b=3 a^5-b^5=781

The $n^{th}$ term is given as

$4^{n-1}-3^{n-1}$ .

Therefore, the $6^{th}$ term is

$4^{6-1}-3^{6-1}=4^{5}-3^{5}=781$ .

First term tells that starting index is 0. Second term (with index 1) tells that a and b are consecutive integers. (a > b) Third term tells that a = 4, b = 3. Thus, 4^5 - 3^5 = 781