Question

an incomplete table of sequences and common ratios is shown.

Sequence #1: 3, 6, 12, 24, 48,..... what is the common ratio of this set

Sequence #2: 15, 9, 27/5, 81/25, 243/125... what is the common ratio of this set?

Answer 1

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

### Sequence #1: 3, 6, 12, 24, 48,...

To find the common ratio (r):
- \( r = \frac{6}{3} = 2 \)
- \( r = \frac{12}{6} = 2 \)
- \( r = \frac{24}{12} = 2 \)
- \( r = \frac{48}{24} = 2 \)

The common ratio for Sequence #1 is **2**.

### Sequence #2: 15, 9, \(\frac{27}{5}\), \(\frac{81}{25}\), \(\frac{243}{125}\)...

To find the common ratio (r):
- \( r = \frac{9}{15} = \frac{3}{5} \)
- \( r = \frac{\frac{27}{5}}{9} = \frac{27}{5} \cdot \frac{1}{9} = \frac{27}{45} = \frac{3}{5} \)
- \( r = \frac{\frac{81}{25}}{\frac{27}{5}} = \frac{81}{25} \cdot \frac{5}{27} = \frac{405}{675} = \frac{3}{5} \)
- \( r = \frac{\frac{243}{125}}{\frac{81}{25}} = \frac{243}{125} \cdot \frac{25}{81} = \frac{6075}{10125} = \frac{3}{5} \)

The common ratio for Sequence #2 is **\(\frac{3}{5}\)**.