Final answer:
To determine the next term in the sequence, identify a pattern in the ratios between the terms. It appears that after the first jump, the sequence multiples by 1.5 for each following term. Using this, the next term after 729 is 729 multiplied by 1.5, which is 1093.5, rounded to the nearest whole number, 1094.
Step-by-step explanation:
To find the next number in the geometric sequence, we first need to identify the common ratio of the sequence. Let's look at the given numbers: 243, 324, 486, and 729. To find the common ratio, we divide each term by the one before it:
- 324 ÷ 243 = 1.333...
- 486 ÷ 324 = 1.5
- 729 ÷ 486 = 1.5
We can see that the sequence is not consistent with its ratio after the first term. However, if we look closely, we can consider an alternative which is finding the sequence of ratios and identifying the pattern:
- 243: No preceding number.
- 324 = 243 × (1 + 1/3)
- 486 = 324 × (1 + 1/2)
- 729 = 486 × (1 + 1/2)
Therefore the next term should also be multiplied by the common ratio of 1.5 (since the sequence seems to stabilize at that ratio after the initial term). So:
Next term = 729 × 1.5 = 1093.5
However, because the terms of the sequence are integers, we should consider the next term to be the nearest whole number:
Next term = 1094