Question

Write out the sums. (You do not need to evaluate them.) \(\sum_{n=1}^{5}(-1)^{n-1} 2^{-n}\)

Short answer

Question: Write the sum of $$\sum_{n=1}^{5}(-1)^{n-1} 2^{-n}$$ without performing the calculation. Answer: The sum is $$(-1)^{1-1} 2^{-1}+(-1)^{2-1} 2^{-2}+(-1)^{3-1} 2^{-3}+(-1)^{4-1} 2^{-4}+(-1)^{5-1} 2^{-5}$$ when the values of n are substituted from 1 to 5.

Step-by-step solution

Substitute the values of n from 1 to 5 into the sum formula

The given sum formula is: $$\sum_{n=1}^{5}(-1)^{n-1} 2^{-n}$$ We will substitute n with its values ranging from 1 to 5 (inclusive) in the formula.

Writing out the sum

After substituting the values of n from 1 to 5, we get the following sum: $$(-1)^{1-1} 2^{-1}+(-1)^{2-1} 2^{-2}+(-1)^{3-1} 2^{-3}+(-1)^{4-1} 2^{-4}+(-1)^{5-1} 2^{-5}$$

Key concepts

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, we often use letters to represent numbers. These letters are called variables. Algebra simplifies complex problems by using these symbols to stand in for unknown values. This makes it easier to write and solve equations. By solving algebraic equations, you can find the values of unknown variables.
When working with sigma notation, algebraic expressions can be used to represent a series of terms concisely. The sigma symbol (\( \Sigma \)) is used to denote the sum of a sequence of terms, allowing you to express lengthy sums more succinctly. Becoming comfortable with algebra and its principles, including understanding how to manipulate variables and expressions, is vital to solving higher-level math problems effectively.

Exponents

Exponents are a way to express repeated multiplication of the same number. For example, \(2^3\) means \(2 \times 2 \times 2\), which equals 8. In the expression \(2^n\), \(2\) is the base, and \(n\) is the exponent. Exponents are a key part of our exercise, especially when you deal with sequences. Knowing how to read and solve them can help you understand and write series much more simply.
Exponents follow specific rules, known as the laws of exponents, which make calculations easier. Some of these include:

• Multiplication: \(a^m \times a^n = a^{m+n}\)
• Division: \(a^m \div a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)

In sigma notation, exponents can be used to simplify the computation of large series of powers efficiently. Understanding how to apply these rules in the context of series allows you to sum up terms without multiplying each pair manually.

Sequences and Series

Sequences are ordered lists of numbers following a particular pattern, and a series is the sum of the terms of a sequence. In our case, we are dealing with a series represented using sigma notation. Sigma notation provides a shorthand way of writing complex series by specifying just the starting and ending values (\(n=1\) to \(n=5\)), and a general rule that describes the relationship of the terms, like \((-1)^{n-1} 2^{-n}\).
Understanding the distinction between sequences and series is crucial:

• A sequence is a list: \(a_1, a_2, a_3, \ldots, a_n\)
• A series is a sum: \(a_1 + a_2 + a_3 + \ldots + a_n\)

In the given exercise, terms are generated by substituting values into the rule, resulting in a list of elements: \(2^{-1}, -2^{-2}, 2^{-3}, -2^{-4}, 2^{-5}\). This helps consolidate understanding of how series are constructed and why sigma notation becomes particularly useful, as it simplifies writing and calculating terms for large sums.