My brother gave me a series to find the next number
11, 42, 93, 164, 255, 366, 497, 648, __.
The logic while making the sum was to append the square of the units digit to the left of the unit digit.
Example square of 1 is appended to 1 hence 11, square of 2 is left appended to 2 so 42...
When I was solving this I found that the series was following a pattern. When you subtract 2 consecutive numbers you get 31, 51, 71, 91, 101, 111, 131 etc.
I am just curious as to why the difference is increasing by 20. It would be really great if someone can give some explanation for this. Is it just by luck or something happening here?
NOTE: This series follows this pattern only till units digit as 9. After that the pattern of difference that I found breakes.
For single digit $n$, the term is $10n^2+n$.
So the difference between successive terms is $10(n+1)^2+(n+1) -(10n^2+n) = 20n+11$ as in the pattern you have spotted.
If $n$ has two digits then the term is $100n^2+n$ with differences of $200n+101$ with a slight glitch when $n=9$ and $n+1=10$. And similarly for larger $n$.
See OEIS A055436, A055437, A055438 for the three sequences