How do you find ${n^{th}}$ term rule for $375, - 75,15, - 3,...$ ?

Hint: The given sequence is a Geometric Progression. We check the common ratio of the sequence and also find the first term. Then by applying the formula of the general term, i.e., the ${n^{th}}$ term, we evaluate the required term of the given sequence.

Formula used: If $a$ is the first term and $r$ is the common ratio of G.P., then ${a_n} = a{r^{n - 1}}$ .

Complete step by step answer:
The given sequence is $375, - 75,15, - 3,...$
We first check the ratio of the terms by dividing the second term by the first term to check the sequence is a G.P.
Here, $a = 375$ and we denote the common ratio as $r$ .
Now, $r = \dfrac{{ - 75}}{{375}} = \dfrac{{ - 1}}{5}$ ,
$r$ is calculated by dividing the second term by the first term.
Thus we find that the common ratio is $\dfrac{{ - 1}}{5}$ .
Then, as the ratio of the terms in the sequence is the same, this means that the given sequence is a geometric progression. Also, abbreviated as, G.P.
So, as we know the general term or ${n^{th}}$ term is
${a_n} = a{r^{n - 1}}$
$\Rightarrow {a_n} = 375.{\left( {\dfrac{{ - 1}}{5}} \right)^{n - 1}}$ ,
We substitute the value of the first term and common ratio.
$\Rightarrow {a_n} = 3 \times {5^3}.{\left( {\dfrac{{ - 1}}{5}} \right)^{n - 1}}$ ,
We write $\;375$ as multiple of its factors
$\Rightarrow {a_n} = 3 \times \dfrac{{{5^3}}}{{{5^{n - 1}}}}.{\left( { - 1} \right)^{n - 1}}$ ,
Here, we write ${5^{n - 1}}$ separately,
$\Rightarrow {a_n} = 3 \times {5^{3 - (n - 1)}}.{\left( { - 1} \right)^{n - 1}}$,
Here we used $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$
$\Rightarrow {a_n} = 3 \times {5^{3 - n + 1}}.{\left( { - 1} \right)^{n - 1}}$
By simplifying the exponent, we get
$\Rightarrow {a_n} = 3 \times {5^{4 - n}}.{\left( { - 1} \right)^{n - 1}}$

So, the ${n^{th}}$ term rule of the sequence is ${a_n} = 3{\left( { - 1} \right)^{n - 1}}{5^{4 - n}}$ .

Note: The common ratio of the terms in the sequence must be constant, then only the sequence is called the geometric progression. In the general term of a geometric progression, $n$ will remain as it is. Also, by using the general term, we can find any other required term of the sequence by substituting the number of terms at the place of $n$ .