One solution is to consider different number bases. Adding 1 to a 7-digit number in one base equals a 9-digit number in a different base. If we call the bases a and b (b<a) we can represent a 7-digit number in one base (base a) as a range of integers a6 to a7-1, and a 9-digit number in the other base as b8 to b9-1.
We should be able to find at least one number x such that xb=xa+1.
The lowest base is 2 and the next lowest is 3, so b=2 and a=3. To find out if these bases can be used to find a solution, we need to look at 36 and 37-1, that is, 72910 to 218610; and 28 and 29-1, that is, 25610 to 51110.
These two ranges don't overlap, so we can't use these bases.
Let's move to a=4 and b=3. This time we have 46 and 47-1, a range 409610 to 1638310; and 38 and 39-1, b range 656110 to 1968210. These ranges overlap between 656110 and 1638310, so x can be any number in the range 6561≤x≤16383. If we take the central number x = (6561+16383)/2=11472, we need to convert this number into the bases: 11472=23031004 (7 digits) and 1202012203 (9 digits).
Now add 1 to the 7-digit number: 23031014 and this becomes the 9-digit number: 1202012213.
23031014=2×4096+3×1024+3×64+16+1=8192+3072+192+16+1=1147310.
1202012213=6561+2×2187+2×243+27+2×9+2×3+1=6561+4374+486+27+18+6+1=1147310.