Problem 43 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{5}\left(3^{i}-4\right)$$

Step-by-Step Solution

Answer

The sum of the series is 343.

Step 1: Identify the Series Terms

The series given is \( \sum_{i=1}^{5} (3^i - 4) \). We need to identify each term from \( i = 1 \) to \( i = 5 \). These terms are: \( 3^1 - 4 \), \( 3^2 - 4 \), \( 3^3 - 4 \), \( 3^4 - 4 \), and \( 3^5 - 4 \).

Step 2: Calculate Each Term in the Series

Now, calculate each individual term:- For \( i = 1 \), \( 3^1 - 4 = 3 - 4 = -1 \).- For \( i = 2 \), \( 3^2 - 4 = 9 - 4 = 5 \).- For \( i = 3 \), \( 3^3 - 4 = 27 - 4 = 23 \).- For \( i = 4 \), \( 3^4 - 4 = 81 - 4 = 77 \).- For \( i = 5 \), \( 3^5 - 4 = 243 - 4 = 239 \).

Step 3: Add All Terms to Find the Sum

Add up all the calculated terms from Step 2:\(-1 + 5 + 23 + 77 + 239\).Calculate the sum:\(-1 + 5 = 4\)\(4 + 23 = 27\)\(27 + 77 = 104\)\(104 + 239 = 343\)

Key Concepts

Arithmetic Series

An arithmetic series is a series of numbers in which each term increases by a constant difference. These series are particularly important in the study of mathematics because they allow us to calculate long sequences quickly.

• For example, the series 2, 4, 6, 8 is an arithmetic series where each number increases by 2.
• The nth term can be calculated using the formula: \( a_n = a + (n-1) \cdot d \), where \( a \) is the first term and \( d \) is the common difference.

In the exercise, the given series is not an arithmetic series but rather involves exponential functions subtracted by a constant. Understanding arithmetic series, however, provides a foundation for recognizing different types of series we might encounter.

Exponential Functions

Exponential functions form a cornerstone of mathematics, often written as \( f(x) = a^x \) where \( a \) is a constant and \( x \) is an exponent. These functions grow at increasingly rapid rates as the exponent increases. In our exercise, the terms in the series include the exponential function \( 3^i \), where the base 3 is repeatedly raised to successive powers from 1 to 5.

• Exponential growth occurs when the rate of change of a quantity is proportional to the current amount of that quantity.
• Examples are population growth, bank interest calculations, and radioactive decay.

By incorporating an exponential function in the series, we see how quickly the terms grow. For instance, in the step-by-step solution, we identified the exponential growth from 3, 9, 27, 81 to 243 as the power increases from 1 to 5.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are fundamental in forming equations and solving problems.

• An example of an algebraic expression is \( 3^i - 4 \), where \( 3^i \) is the exponential expression and \( -4 \) is a constant term.
• Algebraic expressions can be simplified or manipulated to find unknown values by applying algebraic techniques.

In the exercise, each term in the series is an algebraic expression composed of exponential and constant parts. Understanding these expressions is crucial for calculating the overall sum of the series, as demonstrated when we calculated each term individually and summed them up.