Problem 1 - Applied Calculus

Question

Find the sum of the following series in two ways: by adding terms and by using the geometric series formula. $$ 3+3 \cdot 2+3 \cdot 2^{2} $$

Step-by-Step Solution

Answer

The sum of the series is 21.

Step 1: Identify the Terms of the Series

The series given is \(3 + 3 \cdot 2 + 3 \cdot 2^2\). This consists of three terms: \(3\), \(3 \cdot 2\) (which is 6), and \(3 \cdot 2^2\) (which is 12).

Step 2: Add the Terms Directly

Add up the terms from the series. Perform the addition: \(3 + 6 + 12 = 21\).

Step 3: Recognize as a Geometric Series

The series is a geometric series with the first term \(a = 3\) and common ratio \(r = 2\). It has 3 terms.

Step 4: Use the Geometric Series Formula

The formula for the sum \(S_n\) of a geometric series is: \[S_n = a \frac{r^n - 1}{r - 1}\]Substituting the values, where \(n = 3\), \(a = 3\), and \(r = 2\):\[S_3 = 3 \frac{2^3 - 1}{2 - 1} = 3 \frac{8 - 1}{1} = 3 \times 7 = 21\]

Step 5: Verify Both Methods Agree

By both direct addition of terms and using the geometric series formula, the sum for the series is 21. This confirms the consistency of the solution.

Key Concepts

Series Summation

When dealing with a series, a common task is to find its total sum. This is also known as the series summation. In our original exercise, we see a series of numbers: \(3 + 6 + 12\).
We can sum these numbers directly by simple addition:

• Add the first number 3 to the second number 6, giving 9.
• Next, add 9 to the third number 12 to reach a total of 21.

To verify this result, it can also be summed using a formula from math, like with a geometric series formula in this case, providing a second method to ensure accuracy. This approach reinforces the correctness of our sum, highlighting the value of mathematical formulas in confirming calculations are consistent and reliable.
Knowing how to sum a series this way is a useful skill, particularly when dealing with more complex sequences.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the exercise, the series \(3 + 3 \cdot 2 + 3 \cdot 2^2\) is a geometric sequence.
Here’s how we can identify key aspects of a geometric sequence:

• The first term \(a = 3\).
• Each subsequent term is obtained by multiplying the previous term by the common ratio \(r = 2\).
• The terms are \(3, 6,\) and \(12\).

Understanding the nature of a geometric sequence helps in applying the geometric series formula effortlessly. The formula consolidates the terms for quick and efficient summation.

Arithmetic Operations

Arithmetic operations are basic mathematical procedures involving addition, subtraction, multiplication, and division. These operations are fundamental in solving various math problems, including summing a series as in our exercise.
To break it down:

• Identify and apply the operation required at each step. In our series, addition is used to sum individual terms.
• Multiplication comes into play when calculating terms in the geometric sequence (like \(3 \times 2\) or \(3 \times 2^2\)).

Being comfortable with these operations is crucial, as they form the foundation of many more complex calculations. Correctly applying arithmetic operations ensures accurate results, whether you're dealing with a simple series or more advanced equations involving geometric series.