Understanding the Sum to Infinity of an Infinite Geometric Sequence
The sum to infinity of a geometric sequence is a fascinating topic in mathematics, and it applies to sequences where each term is a constant multiple of the previous term. For example, the sequence 256, 64, 16, 4 is a geometric sequence, and we will explore how to calculate its sum to infinity.
Introduction to Geometric Sequences
A geometric sequence is defined as a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the sequence 256, 64, 16, 4, the first term (a) is 256, and the common ratio (r) is 0.25, as each subsequent term is 1/4 of the previous term.
Formula for Sum to Infinity
The sum to infinity (S) of a geometric sequence, provided that the common ratio |r|
$$ S frac{a}{1 - r} $$
Here, 'a' represents the first term, and 'r' represents the common ratio.
Calculating the Sum
For the given sequence 256, 64, 16, 4, we can use the formula to find the sum to infinity.
First term (a): 256 Common ratio (r): 0.25Plugging these values into the formula:
$$ S frac{256}{1 - 0.25} frac{256}{0.75} 341frac{1}{3} $$
Alternative Interpretations
The sequence can be visualized as a geometric series, which is a series of the form $$ a ar ar^2 ar^3 ... $$. Here, the sum to infinity can be understood as the sum of an infinite geometric series with the first term 256 and the common ratio 1/4:
( S_infty frac{256}{1 - frac{1}{4}} frac{1024}{3} 341frac{1}{3} )
Another way to look at this is to consider the sequence as the sum of two geometric series. The first series is $$ 256, 64, 16, 4 ... $$, and the second series is $$ frac{1}{4}, frac{1}{16}, frac{1}{64}, frac{1}{256} ... $$. The sum of these two series is equivalent to the original sequence, and the sum to infinity is calculated as:
( 2 times frac{1}{1 - frac{1}{4}} 2 times frac{1}{frac{3}{4}} 2 times frac{4}{3} frac{8}{3} 341frac{1}{3} )
Conclusion
The sum to infinity of the geometric sequence 256, 64, 16, 4 is 341 1/3. This method can be applied to other geometric sequences where the common ratio is between -1 and 1. Understanding these concepts is crucial for solving various problems in mathematics and can be particularly useful in fields such as finance, physics, and engineering.