# Question 351d2

Apr 13, 2017

B) will work.

#### Explanation:

B) states: ${a}_{n} = {\left(- 1\right)}^{n} / \left(3 n - 2\right)$

The sequence is: $- 1 , \frac{1}{4} , - \frac{1}{7} , \frac{1}{10} , - \frac{1}{13} , \ldots$

If we look at the numerator (top) of the fractions only, we can see that the function will give us an alternating $- 1 , + 1$ as follows:

When $a = 1$ then ${\left(- 1\right)}^{1} = - 1$ to give us a negative number
When $a = 2$ then ${\left(- 1\right)}^{2} = 1$ to give us a positive number
When $a = 3$ then ${\left(- 1\right)}^{3} = - 1$ to give us a negative number
When $a = 4$ then ${\left(- 1\right)}^{4} = 1$ to give us a positive number

That gives us our alternating signs in the sequence, starting with negative, and it also provides us with a $1$ in every numerator.

If we look at the denominator (bottom) of the fraction only, we can see that the function will give us $3 n - 2$. That means each time we move up another number in the sequence we have to solve for $3 n - 2$.

When $a = 1$ then $3 n - 2 = 3 \left(1\right) - 2 = 1$ to give us $- \frac{1}{1} = 1$
When $a = 2$ then $3 n - 2 = 3 \left(2\right) - 2 = 4$ to give us $\frac{1}{4}$
When $a = 3$ then $3 n - 2 = 3 \left(3\right) - 2 = 7$ to give us $- \frac{1}{7}$
When $a = 4$ then $3 n - 2 = 3 \left(4\right) - 2 = 10$ to give us $\frac{1}{10}$

To check, you can calculate the fifth number in the sequence from the formula:

${a}_{5} = {\left(- 1\right)}^{n} / \left(3 n - 2\right) = {\left(- 1\right)}^{5} / \left(3 \left(5\right) - 2\right) = - \frac{1}{15} - 2 = - \frac{1}{13}$

Apr 13, 2017

Take a look at the sequence and see what is changing from term to term. Pick the sequence that matches these guidelines.

#### Explanation:

We observe these conditions in the sequence:

1. The first term is –1 (which we can think of as –1/1).
2. In each new term, the denominator is going up by 3.
3. The terms alternate between negative and positive.

The correct formula will be the one that satisfies these conditions.

The correct answer CANNOT be A, since the terms it will generate do not alternate in sign. (Its terms will always be negative for any $n > 0$.)

The correct answer CANNOT be C, because when $n = 1$, its first term is (–1)^0/(1+3(0))=1/1, which is not –1. (Its terms will have the opposite sign of those in the given sequence.)

The correct answer CANNOT be D, since when $n = 1$, its first term is (–1)^1/(3(0)+2)=–1/2, which is not –1.

That means the answer should be B. Looking at the formula in B, we see that the terms it will generate...

1. ...begin with –1 (since a_1=(–1)^1/(3(1)-2)=–1/(3-2)=–1),
2. ...will alternate in sign (thanks to the (–1)^n in the numerator), and
3. ...their denominators go up by 3 (because of the coefficient of $3$ on the $n$).

To test this, we try $n = 2$:

a_2=(–1)^2/(3(2)-2)=1/(6-2)=1/4

Similarly, when $n = 3$:

a_3=(–1)^3/(3(3)-2)=(–1)/(9-2)=–1/7#

This is clearly generating the sequence we are trying to match. Thus, B is the answer.