Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Find the sum for each series. $$\sum_{i=1}^{4}\left[(-2)^{i}-3\right]$$

Step-by-Step Solution

Answer

The sum of the series is -2.

Step 1: Write Down the Series Formula

The series is given by the formula \( \sum_{i=1}^{4} \left[ (-2)^i - 3 \right] \). This means we need to substitute each integer value of \( i \) from 1 to 4 into the expression \( (-2)^i - 3 \) and sum the results.

Step 2: Substitute and Calculate Each Term

Let's substitute and compute each term:- For \(i = 1\): \((-2)^1 - 3 = -2 - 3 = -5\)- For \(i = 2\): \((-2)^2 - 3 = 4 - 3 = 1\)- For \(i = 3\): \((-2)^3 - 3 = -8 - 3 = -11\)- For \(i = 4\): \((-2)^4 - 3 = 16 - 3 = 13\)

Step 3: Sum All Calculated Terms

Now sum up all the results from each term: \(-5 + 1 - 11 + 13\)- Start with the initial two terms: \(-5 + 1 = -4\)- Add the result to the next term: \(-4 - 11 = -15\)- Finally, add the last term: \(-15 + 13 = -2\)

Step 4: State the Final Answer

The sum of the series \( \sum_{i=1}^{4} \left[ (-2)^i - 3 \right] \) is \(-2\).

Key Concepts

Understanding Summation Notation

Summation notation provides a clear and concise way to express the addition of a sequence of numbers. It is usually represented using the Greek letter "Σ" (sigma). This notation includes an index of summation, which in this case is "i", and runs from a starting value to an ending value as indicated below the sigma and above it. For the given exercise, the notation \( \sum_{i=1}^{4} \left[ (-2)^i - 3 \right] \) indicates that we will calculate the expression \((-2)^i - 3\) for each integer value of \(i\) starting from 1 up to 4.

The results of these calculations are then summed together to find the total sum of the series. By using summation notation, complex series like these can be written in a much simpler form.

Power of Exponents Explained

Exponents are a shorthand way of indicating repeated multiplication of a number by itself. They consist of a base and an exponent or power. In the context of this exercise, \((-2)^i\) tells us to multiply \(-2\) by itself \(i\) times. For example, \((-2)^1 = -2\), \((-2)^2 = 4\), \((-2)^3 = -8\), and so on.

When the exponent is even, as with \((-2)^2 = 4\), the result will be positive because multiplying two negative numbers results in a positive product. Conversely, when the exponent is odd, the product remains negative, such as with \((-2)^3\). Understanding how exponents affect the value of numbers is crucial in dealing with any expressions or series involving powers.

Applying Integer Substitution

Integer substitution is the process of replacing a variable, in this case "i", with actual integer values. This allows us to evaluate expressions or series precisely. For this exercise, the variable \(i\) takes each integer value from 1 to 4 sequentially.

You begin by substituting \(i = 1\) into the expression \((-2)^i - 3\), which gives \((-2)^1 - 3 = -5\). Then, proceed with \(i = 2\) resulting in \((-2)^2 - 3 = 1\), and so on. By substituting each value one at a time, you can evaluate the expression for each specific \(i\). Finally, these individual results are added together to obtain the series' total sum.

Step-by-Step Solution

Understanding and systematically following a step-by-step approach to problems can simplify complex calculations and ensure accuracy. Start by clearly writing down the series formula, as was done in the exercise: \( \sum_{i=1}^{4} \left[ (-2)^i - 3 \right] \).

Next, substitute integer values of \(i\) into the expression and compute each term individually, as outlined in the solution. This involves calculating for \(i = 1, 2, 3,\) and \(4\). After computing, sum all the calculated terms in sequence, breaking it down into manageable parts to avoid errors.

Finally, conclude by stating the sum of the series, like this particular example, gives \(-2\). Following a step-by-step approach helps streamline problem-solving and ensures that all steps are followed correctly, leading to a precise result.