Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{j=1}^{6}-(3.6)^{j}$$

Step-by-Step Solution

Answer

The sum is approximately -3012.622.

Step 1: Understanding the Summation Notation

The given expression is a summation notation, which indicates that you need to add the values of \(-(3.6)^j\) for each whole number value of \(j\) from 1 to 6. This means you substitute \(j = 1\), \(j = 2\), \(j = 3\), and so on, up to \(j = 6\).

Step 2: Substitute the Values of j

Calculate the value of \(-(3.6)^j\) for each \(j = 1, 2, 3, 4, 5, 6\):- For \(j = 1\), compute \(-(3.6)^1\).- For \(j = 2\), compute \(-(3.6)^2\).- Continue similarly until \(j = 6\).Ensure each individual term is calculated accurately.

Step 3: Calculate Each Term

Perform the calculations:- \(j = 1\): \(-(3.6)^1 = -3.6\)- \(j = 2\): \(-(3.6)^2 = -12.96\)- \(j = 3\): \(-(3.6)^3 = -46.656\)- \(j = 4\): \(-(3.6)^4 = -167.9616\)- \(j = 5\): \(-(3.6)^5 = -604.66176\)- \(j = 6\): \(-(3.6)^6 = -2176.782336\)Round each of these values to the nearest thousandth if needed.

Step 4: Sum the Calculated Values

Add the values calculated in the previous step:\[-3.6 - 12.96 - 46.656 - 167.9616 - 604.66176 - 2176.782336\]

Step 5: Use a Graphing Calculator

Enter the series directly into a graphing calculator using the summation function if available, or individually enter and sum the values calculated to get the result. Round the sum to the nearest thousandth.

Step 6: Final Calculation and Rounding

Using the graphing calculator, the sum is approximately \(-3012.621696\). Rounding to the nearest thousandth gives \(-3012.622\).

Key Concepts

Exponents

Exponents are a mathematical way to express repeated multiplication of the same number by itself. For instance, when you see something like \((3.6)^j\), it means the number 3.6 is being multiplied by itself \(j\) times.

• \(3.6^1\) simply equals 3.6.
• \(3.6^2\) is the product of 3.6 multiplied by 3.6.
• \(3.6^3\) means you multiply 3.6 by itself three times.

Exponents are crucial in mathematics because they allow us to simplify expressions and calculations. They can sometimes be intimidating at first glance, but remember that they just represent a form of shorthand for multiplication.Negative exponents might be used as well. In our context, we calculated \(-(3.6)^j\) for values from 1 to 6. This negative sign doesn't change the multiplication process—it just changes the final result to a negative value, as seen in the expressions.Understanding how exponents work is essential, as they are commonly used in various mathematical disciplines and real-world applications. With practice, handling them becomes intuitive.

Graphing Calculator

A graphing calculator is a powerful tool that can greatly assist in solving complex mathematical problems. Unlike a basic calculator, it can plot graphs, solve equations, and evaluate summations, like the one in our exercise.When dealing with summation notation, like \[ \sum_{j=1}^{6} -(3.6)^j \] a graphing calculator can quickly perform these calculations for you.

• First, access the summation function, which allows you to input the limits and expression for evaluating the series.
• Enter the expression \(- (3.6)^j \) directly, setting the lower limit as 1 and the upper limit as 6.
• The calculator computes each term, applies any necessary arithmetic operations, sums them up, and provides a direct result.

The use of a graphing calculator makes handling exponents and summations much simpler and less error-prone, especially when multiple terms are involved. Moreover, this also helps ensure accuracy when results need to be rounded to a certain place value, like the nearest thousandth.

Rounding Numbers

Rounding numbers simplifies them to a specified degree of precision, making them easier to work with. In mathematics, rounding involves modifying a number so that it is a closer approximation by keeping the significant digits and adjusting the rest.When rounding to the nearest thousandth, the number should have three decimal places retained. For example, if you encounter a number like \(-3012.621696\), following these steps will help you round correctly:

• Look at the fourth digit after the decimal point—in this case, it is 6.
• If this digit is 5 or above, increase the third digit by 1. For \(-3012.621696\), increasing the third digit means it becomes 2 + 1 = 3.
• Thus, the rounded number becomes \(-3012.622\).

Rounding is an essential skill in math, as it provides quick approximations that are easier to handle in everyday calculations and reporting. It helps maintain precision without getting overwhelmed by too many decimal places, balancing accuracy and practicality.