Problem 25 - Algebra 2

Question

Tell whether each list is a sequence or a series. Then tell whether it is finite or infinite. $$ 1,2,4,8,16,32, \dots $$

Step-by-Step Solution

Answer

The given list is an infinite sequence.

Step 1: Determine Sequence or Series

The list given: 1,2,4,8,16,32, \ldots presents a pattern where each number is double the previous number, showing it's a sequence, not a sum which would define it as a series.

Step 2: Determine if Finite or Infinite

The list ends with \ldots indicating that it continues indefinitely. Thus, it is an infinite sequence rather than a finite set.

Key Concepts

Infinite Sequence

An infinite sequence is a sequence that continues indefinitely without terminating. Unlike finite sequences, infinite sequences do not have a final term. In this context, the list given in the exercise starts with 1, 2, 4, 8, 16, 32, and continues indefinitely, as indicated by the "...". This shows that it doesn't have an end point.

Infinite sequences are represented mathematically with a general formula for the nth term, which helps to understand the pattern they follow. For example, in an arithmetic sequence where each term increases by the same difference, the nth term can be defined by a simple formula. However, this exercise shows a geometric sequence, which follows multiplication by a constant to get to the next term. Therefore, infinite sequences aid in comprehending larger mathematical concepts and are fundamental in understanding series, limits, and calculus.

Geometric Progression

A geometric progression, or geometric sequence, is a type of sequence where each term is found by multiplying the previous one by a constant number, called the ratio. In the given exercise, the sequence presented is an example of a geometric progression as each term is double the previous one, meaning the common ratio is 2.

The general formula for a geometric sequence can be expressed as:

• \( a_n = a_1 imes r^{n-1} \)

where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio. Here, \( a_1 = 1 \) and the ratio \( r = 2 \). By applying this formula, one can find any term in the sequence, including any theoretical term as the sequence progresses indefinitely. Geometric progressions are powerful in modeling exponential growth or decay and are widely used in various scientific and engineering applications.

Finite Sequence

A finite sequence has a specific number of terms and comes to an end, unlike an infinite sequence. Each element in the sequence can be counted, and there is a clear beginning and a definitive last term.

For instance, if our sequence stopped at 32 instead of continuing with ",...", it would be an example of a finite sequence consisting of the numbers 1, 2, 4, 8, 16, and 32. Finite sequences are frequently encountered in everyday activities and are critical in various disciplines, including mathematics for solving problems and making calculations more manageable.

In contrast to the infinite sequence discussed earlier, a finite sequence can often be evaluated and summed easily since the number of terms is limited and known. Finite sequences often form the foundation for developing and understanding broader mathematical concepts such as series, particularly when we are interested in summing their terms.