Question

in a geometric sequence, the ratio of consecutive terms is constant. the constant ratio is called the common ratio, often represented by r. in the above example, the common ratio is 3.

the common ratio can be used to find additional terms. simply multiply the common ratio by any term to obtain the next term. for the sequence 5, 15, 45, 135, 405... we can simply multiply 405 by 3 to obtain which is the next term.

follow the steps listed to find the common ratio of the sequence: 3, 6, 12, 24, 48...

a complete each division.
\\(\frac{6}{3} = \\)
\\(\frac{12}{6} = \\)
\\(\frac{24}{12} = \\)
\\(\frac{48}{24} = \\)

b hence, the common ratio is ______.

c use the common ratio you found to identify the next term in the geometric sequence:
the next term is \\(48 \cdot \\) ____ = ____.

each term of a geometric sequence is the product of the preceding term and the common ratio r.

Explanation:

Calculate consecutive term ratios

To find the common ratio, we divide each term by its preceding term.

$$ \frac{6}{3} = 2 $$

$$ \frac{12}{6} = 2 $$

$$ \frac{24}{12} = 2 $$

$$ \frac{48}{24} = 2 $$

Identify the common ratio

Since each division yields the same constant value, the common ratio \(r\) is:

$$ r = 2 $$

Calculate the next term

To find the next term, multiply the last term by the common ratio.

$$ 48 \cdot 2 = 96 $$

Answer:

Question A

$$\frac{6}{3} = 2$$

$$\frac{12}{6} = 2$$

$$\frac{24}{12} = 2$$

$$\frac{48}{24} = 2$$

Question B

Hence, the common ratio is 2.

Question C

The next term is \(48 \cdot 2 = 96\).