Is the following an arithmetic sequence, geometric sequence or neither: 324, 108, 36, 12, 4, …
The Correct Answer and Explanation is:
To determine whether the sequence 324, 108, 36, 12, 4 is an arithmetic sequence, a geometric sequence, or neither, we need to analyze the pattern of the numbers.
- Arithmetic Sequence: An arithmetic sequence is defined as a sequence in which the difference between consecutive terms is constant. To check if our sequence is arithmetic, we will calculate the differences between each pair of consecutive terms:
- Difference between 324 and 108: ( 108 – 324 = -216 )
- Difference between 108 and 36: ( 36 – 108 = -72 )
- Difference between 36 and 12: ( 12 – 36 = -24 )
- Difference between 12 and 4: ( 4 – 12 = -8 ) The differences are: -216, -72, -24, and -8. Since the differences are not constant (they are decreasing and not equal), the sequence is not an arithmetic sequence.
- Geometric Sequence: A geometric sequence is defined as a sequence where the ratio between consecutive terms is constant. We can find the ratios of consecutive terms in our sequence:
- Ratio between 324 and 108: ( \frac{108}{324} = \frac{1}{3} )
- Ratio between 108 and 36: ( \frac{36}{108} = \frac{1}{3} )
- Ratio between 36 and 12: ( \frac{12}{36} = \frac{1}{3} )
- Ratio between 12 and 4: ( \frac{4}{12} = \frac{1}{3} ) The ratios are all equal to ( \frac{1}{3} ). Since the ratio between each pair of consecutive terms is constant, this sequence is classified as a geometric sequence.
In conclusion, the sequence 324, 108, 36, 12, 4 is a geometric sequence with a common ratio of ( \frac{1}{3} ). The constant ratio confirms that each term is obtained by multiplying the previous term by ( \frac{1}{3} ). Thus, the correct answer is that it is a geometric sequence.