In the question i'm asked to compute the compression rate in LZ algorithm of a string of the form "abc"*k for 0<k<10

Its not clear to me what kind of answer is expected. There doesn't seem to be a closed expression for this value as a function of k.

The intermediate forms look like this:

k=1:

[a,b,c]

k=2:

[a,b,c,(3,3)]

k=3:

[a,b,c,(3,6)]

k=4:

[a,b,c,(3,6),(3,3)]

k=5:

[a,b,c,(3,6),(3,6)]

k=6:

[a,b,c,(3,6),(6,9)]

k=7:

[a,b,c,(3,6),(6,12)]

k=8:

[a,b,c,(3,6),(9,15)]

k=9:

[a,b,c,(3,6),(9,18)]

Was this what i was supposed to do and then calculate the compression rate for each one? (clearly a posteriori its not so difficult because it doesn' change a lot with k but is there a way to see for example the jump from k=3 to k=4 before calculating?)