limits - Math Sequence help

I have a sequence:
7.537803
30.1512
67.8402
120.60
... it keeps going infinitely. There seems to be no equation to it. I have noticed however, that the ratio between subsequent terms tends towards 1:
3.999998408

2.25
1.777780726
...etc
Again, no equation to it. However, my problem here is that whilst the sequence of ratios converges to 1, this should mean that there should eventually be a point where the difference is negligible. However, using massive numbers, I get a negligible difference. Say I use 999,999 and 1,000,000 to calculate results, but then at 999,999,999 and 1,000,000,000 the difference between the results gotten using each number is negligible but the difference between the result for 1,000,000,000 and that for 1,000,000 is quite a lot. If the ratio between subsequent terms converges to 1, and its pretty close to 1 when I use 1,000,000 and 999,999 to calculate a result, shouldnt the results calculated for 999,999,999 and 1,000,000,000 be far closer to that found for 999,999 and 1,000,000 (I say that instead of those because really the difference was very negligible).

Secondly, if I want to say that an equation for a term of another sequence is the previous term+15 how would i write it in mathematical notation?
Thanks!
Update: Also, here's background info if you think it might be helpful.
https://www.desmos.com/calculator/qxmzxx...
Update 2: also-sorry just in case not clear: the difference between 999,999 and 1,000,000 and that between 999,999,999 and 1,000,000,000 was negligible my concern is with the difference between the two groups of numbers if that makes sense

Answer

First question: The term ratio appears to be $\frac{2^2}{1^2}, \frac{3^2}{2^2}, \frac{4^2}{3^2}, \ldots$. So you can imagine, each term is a multiple of the first term, where the multiplier is $n^2$:

$$a_n = a_1n^2 \approx 7.537803 n^2$$

The term difference, $a_{n+1}-a_n$, is
$$\begin{align*}a_{n+1}-a_n
&= a_1[(n+1)^2-n^2]\\
&= a_1(2n+1)\\
&\to \infty

\end{align*}$$

which means the term difference is actually increasing and diverges.

The term ratio, $\frac{a_{n+1}}{a_n}$, on the other hand, is
$$\begin{align*}
\frac{a_{n+1}}{a_n} &= \frac{a_1(n+1)^2}{a_1n^2}\\
&=\frac{n^2+2n+1}{n^2}\\
&= 1+\frac{2n+1}{n^2}\\
&\to 1

\end{align*}$$
which, as you expect, tends to $1$.