Question

consider the sequence.
729, 243, 81, 27, ...
enter the answer in each space provided. use numbers instead of words.
starting number
common ratio

Explanation:

Step1: Identify Starting Number

The sequence starts with 729, so the starting number is 729.

Step2: Calculate Common Ratio

To find the common ratio \( r \) in a geometric sequence, divide a term by its previous term. For example, \( \frac{243}{729} = \frac{1}{3} \), or \( \frac{81}{243} = \frac{1}{3} \), so the common ratio is \( \frac{1}{3} \) (or 0.333... but as a fraction, it's \( \frac{1}{3} \), and as a decimal for the box, we can write it as \( \frac{1}{3} \) or 0.333... but since the problem says use numbers, and the ratio here is \( \frac{1}{3} \) which is 0.333... but more accurately, \( \frac{1}{3} \) is the exact value. Wait, let's check again: 729, 243, 81, 27... 729 divided by 3 is 243, 243 divided by 3 is 81, 81 divided by 3 is 27. Wait, no: 243 is 729 (1/3), 81 is 243 (1/3), so the common ratio is \( \frac{1}{3} \), which is 0.333... but as a fraction, it's \( \frac{1}{3} \). But the problem says "use numbers instead of words", so we can write the common ratio as \( \frac{1}{3} \) or 0.333..., but let's see: 729, 243 (729/3), 81 (243/3), 27 (81/3), so the ratio is 1/3. So starting number is 729, common ratio is 1/3 (or 0.333... but as a fraction, 1/3 is better).

Answer:

Starting Number: 729
Common Ratio: \( \frac{1}{3} \) (or 0.333... but 1/3 is exact)

Wait, but the problem has two boxes. So for the first box (Starting Number), it's 729. For the second box (Common Ratio), it's \( \frac{1}{3} \) (or 0.333... but as a fraction, 1/3 is correct). So:

Starting Number: \(\boxed{729}\)
Common Ratio: \(\boxed{\frac{1}{3}}\) (or if decimal is preferred, \(\boxed{0.333...}\) but 1/3 is exact)