Find general term of $1+\frac{2!}{3}+\frac{3!}{11}+\frac{4!}{43}+\frac{5!}{171}+....$

Hint

The numerator is easy.

For the denominator, see the succesive differences. If you still can't figure out see Arithmetico–geometric sequence.

Notice that the numerator comprises factorials increasing by $1$ in each successive term. For the denominator it requires a bit more observation. The difference between the denominators of successive terms is our cue to guess there's an exponential term involved.

First Guess: $2^{2n+1}+1$ because have a look at the differences, they are differences of $2, 8, 32, \ldots$.

But clearly doing so gives us the denominators as $3, 9, 33,\ldots$ which is thrice of what our actual denominators are so we divide by $3$ to get the desired general $n^{\text{th}}$ term of the sequence.

$$S_n=\sum_{k=0}^{n}\dfrac{3(k+1)!}{2^{2k+1}+1}$$