Problem 40 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=-1}^{2} 5(2)^{i}$$

Step-by-Step Solution

Answer

The sum of the series is 37.5.

Step 1: Understand the Series

The series is represented by the sum of terms from the index \(i = -1\) to \(i = 2\). It is written as \(\sum_{i=-1}^{2} 5(2)^{i}\). This means we need to compute the value of \(5(2)^{i}\) for each integer \(i\) between \(-1\) and \(2\), then sum these values.

Step 2: Calculate Each Term

Calculate the value of \(5(2)^{i}\) for each index:- For \(i = -1\), the term is \(5(2)^{-1} = 5 \cdot \frac{1}{2} = 2.5\).- For \(i = 0\), the term is \(5(2)^{0} = 5 \cdot 1 = 5\).- For \(i = 1\), the term is \(5(2)^{1} = 5 \cdot 2 = 10\).- For \(i = 2\), the term is \(5(2)^{2} = 5 \cdot 4 = 20\).

Step 3: Sum the Terms

Add the calculated terms together:- Sum = \(2.5 + 5 + 10 + 20\).- Therefore, \(\text{Sum} = 37.5\).

Key Concepts

Summation Notation

Summation notation is a concise and convenient way to express the sum of a sequence of terms. It's commonly represented by the Greek letter sigma (\(\Sigma\)). In the exercise given, the notation \(\sum_{i=-1}^{2} 5(2)^{i}\) represents an arithmetic task where we calculate the sum of a sequence of numbers generated by a given formula, starting from a lower limit \(i = -1\) to an upper limit \(i = 2\).
Understanding the components of summation notation is crucial:

• The expression under the sigma symbol, \(5(2)^i\) in this case, tells you how to compute each term in the series.
• The number below the sigma, \(i = -1\), specifies the starting value of the index \(i\).
• The number above the sigma, \(2\), indicates that the index will increase up to 2.

Understanding these elements helps in efficiently breaking down and calculating series, especially when dealing with sequences that follow a mathematical pattern.

Geometric Progression

A geometric progression (or sequence) is a series 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 the given exercise, the expression \(5(2)^i\) represents a form of geometric progression.
To recognize it:

• The fixed number that each term is multiplied by is the base of the exponent, noted in \(2^i\). Here, the common ratio is 2.
• The first coefficient, 5, scales each term uniformly across the series, ensuring consistency when we compute the terms for each index.

For the calculation of the sum in this progression over specific indices, you'd compute each term individually and then add them according to the limits specified by the summation notation. This pattern of multiplying by a constant makes geometric sequences both predictable and manageable once you understand the underlying rules.

Index Calculation

Index calculation refers to determining the specific values at each step within a defined range, as specified in a series or sequence. In the context of the exercise, it involves evaluating the expression \(5(2)^i\) for each value of \(i\) from \(-1\) to \(2\).
Here’s how you can do it step-by-step:

• Begin at the starting index, \(i = -1\), and calculate \(5(2)^{-1} = 2.5\).
• Move to the next index value, \(i = 0\), where the calculation is \(5(2)^0 = 5\).
• Continue similarly and compute \(5(2)^1 = 10\) at \(i = 1\).
• Finally, at \(i = 2\), evaluate \(5(2)^2 = 20\).

Once each term has been calculated, summing these values yields the total sum of the series. This systematic approach ensures no terms are skipped, and each index is calculated precisely to yield an accurate series total.