Krdytkyu

My math book explains the concept of geometric series as:
$ar + ar^2 + ar^3 + ldots$ (upto the $n$th term)

and the formula to find the sum upto $n$ terms is:

Case $1$: (when $r<1$)

$S_n = a_1 dfrac{1-r^n}{1-r}$ (note: $a_1$ is the first term of the series)

Case $2$: (when $r>1$)

$S_n = a_1 dfrac{r^n-1}{r-1}$

Now the question is: when any variable is involved in the series meaning if $r=a$ (or any other variable) we always suppose $r>1$, why is that?

For example consider the question below:

Q) Sum upto $n$ terms the following series:
$1 + (a + b) + (a^2 + ab + b^2) + (a^3 + a^2b + ab^2 + b^3) + ldots$

In the above question we take $a>1$. How is that possible?

There should be the term $a$ regardless. You can only assume $r>1$ if the sequence is increasing, and likewise only assume $r<1$ if it is decreasing.

In my experience, it is only relevant to consider $r neq 1$, which would be the sum of an arithmetic sequence (and indeed, the denominator of the final formula is undefined for this value). And indeed, your case 1 and case 2 are formally identical.

Now for your specific exemple, it seems to me that $r=a+b$, and the condition $a>1$ along with $b>0$ is sufficient to ensure $rneq 1$ and apply the result directly.