What is the sixth term of the sequence 90, -30, 10, -10/3, ….?

Options

A. `1//3`
B. 0
C. `-10//27`
D. `-3`

Answer

To find the sixth term of the sequence 90, -30, 10, -10/3, …, we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term of the sequence \( a_1 \) is 90. To find the common ratio \( r \), we can divide the second term by the first term:
\[ r = \frac{a_2}{a_1} = \frac{-30}{90} = -\frac{1}{3} \]

### Step 2: Confirm the common ratio
We can confirm that the common ratio is consistent by checking the ratio of other consecutive terms:
\[ r = \frac{a_3}{a_2} = \frac{10}{-30} = -\frac{1}{3} \]
\[ r = \frac{a_4}{a_3} = \frac{-\frac{10}{3}}{10} = -\frac{1}{3} \]
Since we get the same ratio, we can conclude that the common ratio \( r \) is indeed \( -\frac{1}{3} \).

### Step 3: Use the formula for the nth term of a geometric sequence
The formula for the nth term \( a_n \) of a geometric sequence is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
Here, we need to find the sixth term (\( n = 6 \)):
\[ a_6 = 90 \cdot \left(-\frac{1}{3}\right)^{6-1} = 90 \cdot \left(-\frac{1}{3}\right)^{5} \]

### Step 4: Calculate \( a_6 \)
Now we calculate \( \left(-\frac{1}{3}\right)^{5} \):
\[ \left(-\frac{1}{3}\right)^{5} = -\frac{1}{243} \]
Now substitute this back into the formula:
\[ a_6 = 90 \cdot \left(-\frac{1}{243}\right) = -\frac{90}{243} \]
Simplifying \( -\frac{90}{243} \):
\[ -\frac{90}{243} = -\frac{10}{27} \]

### Final Answer
Thus, the sixth term of the sequence is:
\[ \boxed{-\frac{10}{27}} \]