Problem 111 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}$$

Step-by-Step Solution

Answer

The Series can be expressed using sigma notation as \(\sum_{n=0}^{4} \frac{2^n-1}{2^{n+2}}\). The exact sum can be found using a graphing utility.

Step 1: Pattern Recognition

The first step is to recognize the pattern in the given series. Each term - apart from the first one - is generated by taking half of the previous term and adding 1/8. The terms in the numerator follow the sequence 1, 3, 7, 15, 31. This pattern is consistent with the series \(2^n-1\), considering that n starts from 0. The denominator follows a simple sequence of powers of 2. So, putting these observations together, each term can be expressed as \(\frac{2^n-1}{2^{n+2}}\).

Step 2: Sigma Notation

The next step is to express the series in sigma notation. This means determining a general term that can be used to generate the full sequence. The general term here is \(\frac{2^n-1}{2^{n+2}}\). Because the series has five terms and the pattern identified matches all of them starting from n=0, the series can be expressed as:\(\sum_{n=0}^{4} \frac{2^n-1}{2^{n+2}}\)

Step 3: Finding the Sum

A graphing utility can be used to compute the sum. Use it to input the sigma notation and calculate the sum. The sum of this series can also be found manually, but the exercise specifies the use of a graphing utility.

Key Concepts

Pattern Recognition

Pattern recognition is crucial when dealing with series in mathematics. It allows us to identify how each term in a sequence is generated. In the example given, the numerators of the series follow the sequence 1, 3, 7, 15, 31. A closer observation reveals that each numerator is of the form \(2^n-1\). Here, - \(n\) represents the index of the term starting from zero. - We can see that \(2^0-1 = 1\), \(2^1-1 = 1\), \(2^2-1 = 3\), and so forth.

• The numerator illustrates a pattern based on powers of 2, minus one.
• The denominator simply follows powers of 2: 4, 8, 16, 32, 64, which are \(2^{n+2}\).

Recognizing this pattern allows us to formulate a general term for the series.

Series Sum

Once a pattern is identified, the next step is to find the sum of the series using sigma notation.
Sigma notation is a convenient way to represent a sum of terms. It uses the Greek letter \(\Sigma\) to denote the sum and includes a general term for the sequence.

• For our sequence, the nth term can be expressed as \(\frac{2^n-1}{2^{n+2}}\).
• The series given has five terms starting from n=0, which we write as:
  \[\sum_{n=0}^{4} \frac{2^n-1}{2^{n+2}}\]

This compact form is essential for calculations, especially for longer sequences.

Graphing Utility

Graphing utilities are powerful tools that can simplify complex calculations. In this exercise, a graphing utility can quickly compute the sum of a series once it is expressed in sigma notation.
Common graphing utilities, such as graphing calculators or computer software, allow you to input the sigma notation directly.

• This approach provides an efficient way to find the sum without manual computation.
• Such utilities might also visualize the sequence, helping to understand its behavior further.

Using a graphing utility for summation provides not just accuracy but also insights through potential visual representation.

Sequence

In mathematics, a sequence is an ordered list of numbers that often follow a particular pattern. Each number in the list is called a term. Recognizing sequences involves identifying the rule that defines the pattern of the sequence.

• This rule is crucial in describing the relationship between terms and simplifies expressing the sequence.
• The given sequence utilizes powers of 2 as its basis, modified to generate the numerators (\(2^n-1\)) and denominators (\(2^{n+2}\)).

Understanding sequences is foundational for developing and comprehending series and finding their sums.