What are the next 3 numbers in the Geometric sequence 9, 3, 1, 1/3?
The correct Answer and Explanation is:
The given geometric sequence is 9, 3, 1, 1/3. To find the next three numbers, we need to determine the common ratio (r) of the sequence.
Step 1: Identify the common ratio
A geometric sequence has a constant ratio between consecutive terms. To find this common ratio, divide any term by its preceding term:
- From 9 to 3: ( \frac{3}{9} = \frac{1}{3} )
- From 3 to 1: ( \frac{1}{3} )
- From 1 to ( \frac{1}{3} ): ( \frac{1/3}{1} = \frac{1}{3} )
Thus, the common ratio (r) is ( \frac{1}{3} ).
Step 2: Calculate the next three terms
We will multiply each term by the common ratio ( \frac{1}{3} ) to find the next terms:
- The term after ( \frac{1}{3} ) is: ( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} )
- The term after ( \frac{1}{9} ) is: ( \frac{1}{9} \times \frac{1}{3} = \frac{1}{27} )
- The term after ( \frac{1}{27} ) is: ( \frac{1}{27} \times \frac{1}{3} = \frac{1}{81} )
So, the next three numbers in the sequence are ( \frac{1}{9}, \frac{1}{27}, \frac{1}{81} ).
Step 3: Explanation
A geometric sequence progresses by multiplying each term by a fixed value, known as the common ratio. In this sequence, the first term is 9, and the ratio between consecutive terms is ( \frac{1}{3} ). This constant ratio is applied to each term to generate the next. The nature of geometric sequences allows them to either grow or shrink, depending on whether the ratio is greater than or less than 1. In this case, since the ratio ( \frac{1}{3} ) is less than 1, the terms decrease progressively. Geometric sequences are often used in mathematical modeling, finance, and physics due to their predictable nature.