The general term of $\frac{2^2}{5!}-\frac{5^7}{6!}+\frac{8^{12}}{7!}-\frac{11^{17}}{8!}+\cdots$

Question

Can someone help me into finding the general term of this sequence? I have already tried at the denominator and according to my calculations I think it should be $(n+4)!$. $$\frac{2^2}{5!} - \frac{5^7}{6!} + \frac{8^{12}}{7!} - \frac{11^{17}}{8!} + \cdots$$

Answer 1

Let's break it down,

The numerator (minus the exponent) $$2,5,8,11$$ This is basically an AP, $2+3(n-1)=3n-1$.

The exponent $$2,7,12,17$$ This is basically another AP, $2+5(n-1)=5n-3$.

The denominator, yes you are right it's $(n+4)!$

The sign, it's $(+1)$ for even terms and $(-1)$ for odd terms, so it's $(-1)^{n+1}$

Altogether, $$\frac{(-1)^{n+1}(3n-1)^{5n-3}}{(n+4)!}$$ for $n=1,2,3,\ldots$