Understanding the Sequence of 1 1/3 and 1/9 and Its Geometric Series Analysis
The given sequence is: 1, 1/3, 1/9. This sequence exhibits a clear pattern that we will explore in detail, including the identification of the sequence type, key mathematical properties, and the calculation of the sum to infinity.
Identifying the Pattern
Let's start by analyzing the sequence: 1, 1/3, 1/9.
Pattern Recognition
The first term, a1, is 1.
The second term, a2, is (1/3^2).
The third term, a3, is (1/3^3).
By observing the sequence, we can see that each term is generated by the formula (a^n 1/3^{(n-1)}), where n starts from 1 and increases by 1 for each subsequent term.
Next Term in the Sequence
Following the pattern, the next term, a4, will be (1/3^4 1/81). Thus, the sequence continues as: 1, 1/3, 1/9, 1/27, 1/81, and so on.
Geometric Sequence Analysis
This sequence is an example of a geometric progression (GP) with a first term of 1 and a common ratio of 1/3. Let's delve into the details of the GP and its properties.
GP Definition and Properties
A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this case, the common ratio is 1/3, which is less than 1, making it a decreasing sequence.
Formulas for GP
The general term of a GP is given by:
[a_n a cdot r^{(n-1)}]Where a is the first term, r is the common ratio, and n is the position of the term in the sequence. For our sequence:
[a_1 1, r 1/3]The formula for the n-th term of the given sequence is:
[a_n 1 cdot (1/3)^{(n-1)} 1/{3^{(n-1)}}]Sum of the Infinite GP
The sum of an infinite geometric sequence, when the common ratio is between -1 and 1 but not equal to 1, is given by:
[S_{infty} frac{a}{1ār}]For our sequence, the sum to infinity (Sā) is:
[S_{infty} frac{1}{1ā1/3} frac{1}{2/3} frac{3}{2}]This means that the sum of the infinite series 1, 1/3, 1/9, 1/27, ... is 1.5 or 3/2.
Practical Example
The next term in the sequence after 1, 1/3, 1/9 is 1/27. Each term in this sequence is found by multiplying the previous term by 1/3.
Conclusion
In conclusion, the given sequence: 1, 1/3, 1/9, ... is a geometric sequence with a first term of 1 and a common ratio of 1/3. The sum to infinity of this sequence is 3/2, and each term is generated by multiplying the previous term by 1/3.
Related Terms
Geometric sequence Geometric progression Sum to infinityUnderstanding these terms and their properties is crucial for analyzing and solving problems involving geometric sequences and series.