Questions: Find the missing terms in the following arithmetic or geometric sequences. a. -2, , 54, - 162 b. - 81 , -9, 3

Solution Steps

To find the missing terms in the sequences, we need to determine whether each sequence is arithmetic or geometric. For an arithmetic sequence, the difference between consecutive terms is constant. For a geometric sequence, the ratio between consecutive terms is constant.

1. Identify the type of sequence (arithmetic or geometric).
2. Calculate the common ratio or difference.
3. Use the common ratio or difference to find the missing terms.

1. Identify the type of sequence (arithmetic or geometric).
2. Calculate the common ratio or difference.
3. Use the common ratio or difference to find the missing terms.

We need to determine if the sequence \(-2, \quad \_, \quad 54, \quad -162\) is arithmetic or geometric.

To check if the sequence is geometric, we calculate the ratio between the known terms: \[ \text{ratio} = \frac{54}{-2} = -27 \] Since the ratio is constant, the sequence is geometric.

Using the common ratio \(-27\), we can find the missing term: \[ \text{missing term} = -2 \times (-27) = 54 \] Thus, the complete sequence is \(-2, \quad 54, \quad 54, \quad -162\).

We need to determine if the sequence \(\_, \quad -81, \quad \_, \quad -9, \quad 3\) is arithmetic or geometric.

To check if the sequence is geometric, we calculate the ratio between the known terms: \[ \text{ratio} = \frac{-9}{-81} = 0.1111 \] Since the ratio is constant, the sequence is geometric.

Using the common ratio \(0.1111\), we can find the missing terms: \[ \text{first missing term} = -81 \div 0.1111 = -729 \] \[ \text{second missing term} = -81 \times 0.1111 = -9 \] Thus, the complete sequence is \(-729, \quad -81, \quad -9, \quad -9, \quad 3\).

Final Answer