What is a formula for the nth term of the given sequence? $$ 10,-20,40 \ldots $$ Answer $$ \begin{array}{ll}a_{n}=-5\left(-\frac{1}{2}\right)^{-n} & a_{n}=10(-2)^{n} \ a_{n}=-5(-2)^{-n} & a_{n}=-5\left(-\frac{1}{2}\right)^{1-n}\end{array} $$

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To find a formula for the $$ n $$th term of the given sequence $$ 10, -20, 40, \ldots $$, we first need to identify the pattern in the sequence.

1. **Identify the terms:**
   - $$ a_1 = 10 $$
   - $$ a_2 = -20 $$
   - $$ a_3 = 40 $$

2. **Calculate the ratios between consecutive terms:**
   - $$ \frac{a_2}{a_1} = \frac{-20}{10} = -2 $$
   - $$ \frac{a_3}{a_2} = \frac{40}{-20} = -2 $$

This indicates that each term is obtained by multiplying the previous term by $$-2$$.

3. **General formula:**
   The first term is $$ 10 $$, and each subsequent term is multiplied by $$-2$$. Therefore, we can express the $$ n $$th term as:
   $$
   a_n = 10 \cdot (-2)^{n-1}
   $$

4. **Check the formula:**
   - For $$ n = 1 $$: $$ a_1 = 10 \cdot (-2)^{1-1} = 10 \cdot 1 = 10 $$
   - For $$ n = 2 $$: $$ a_2 = 10 \cdot (-2)^{2-1} = 10 \cdot (-2) = -20 $$
   - For $$ n = 3 $$: $$ a_3 = 10 \cdot (-2)^{3-1} = 10 \cdot 4 = 40 $$

The formula holds true for the first three terms.

5. **Final answer:**
Thus, the formula for the $$ n $$th term of the sequence is:
$$
a_n = 10 \cdot (-2)^{n-1}
$$

Among the options provided, the equivalent form is:
$$
a_n = 10(-2)^{n}
$$
This is because $$ (-2)^{n-1} $$ can be rewritten as $$ \frac{(-2)^{n}}{-2} $$, which is consistent with the sequence's behavior.

Therefore, the correct formula for the $$ n $$th term of the sequence is:
$$
a_n = 10(-2)^{n-1}
$$
The $$ n $$th term of the sequence is $$ a_n = 10(-2)^{n-1} $$.

What is a formula for the nth term of the given sequence? \[ 10,-20,40 \ldots \] Answer \( \begin{array}{ll}a_{n}=-5\left(-\frac{1}{2}\right)^{-n} & a_{n}=10(-2)^{n} \\ a_{n}=-5(-2)^{-n} & a_{n}=-5\left(-\frac{1}{2}\right)^{1-n}\end{array} \)

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