Problem 40 - Algebra for College Students

Question

$$ \sum_{i=1}^{6} 3^{i} 1092 $$

Step-by-Step Solution

Answer

The sum is 1,192,464.

Step 1: Understand the Exercise

We need to evaluate the summation \( \sum_{i=1}^{6} 3^{i} \cdot 1092 \). This involves finding the sum of the series from \( i = 1 \) to \( i = 6 \) where each term is \( 3^i \cdot 1092 \).

Step 2: Factor Out the Constant

Since \( 1092 \) is constant for each term in the summation, we can factor it out of the summation. This transforms the expression into \( 1092 \cdot \sum_{i=1}^{6} 3^i \).

Step 3: Evaluate the Geometric Series

The sum \( \sum_{i=1}^{6} 3^i \) is a geometric series. Use the formula for the sum of a geometric series \( S_n = a \cdot \frac{r^n - 1}{r - 1} \), where \( a = 3 \), \( r = 3 \), and \( n = 6 \):\[ S_6 = 3 \cdot \frac{3^6 - 1}{3 - 1} \].

Step 4: Calculate Geometric Series Sum

Calculate \( S_6 = 3 \cdot \frac{3^6 - 1}{2} \). First, calculate \( 3^6 = 729 \). Then, substitute into the equation:\[ S_6 = 3 \cdot \frac{729 - 1}{2} = 3 \cdot \frac{728}{2} = 3 \cdot 364 = 1092 \].

Step 5: Compute the Final Result

Substitute the sum of the geometric series \( S_6 = 1092 \) back into the factored expression \( 1092 \cdot S_6 \):\[ 1092 \cdot 1092 = 1192464 \].

Step 6: Conclusion: Verify the Calculation

Review each step to ensure calculations were made correctly. Confirm the use of the geometric series sum formula was appropriate and that all arithmetic operations were performed accurately.

Key Concepts

Summation

Let's dive into the concept of summation, which is a mathematical method for adding together a series of numbers. In the exercise, we are using summation notation, denoted by the Greek letter Sigma (\( \Sigma \)). This notation helps us efficiently express the addition of a series of terms without having to write them all out separately.

In our case, the sum \( \sum_{i=1}^{6} 3^i \cdot 1092 \) involves summing the terms from \( i = 1 \) to \( i = 6 \). This means that we take each power of 3, starting from \( 3^1 \) up to \( 3^6 \), and multiply it by 1092, then add all these products together.

Here is a reminder of what we do in a summation:

• Identify the range of terms from the starting point \( i = 1 \) to the ending point \( i = 6 \).
• Calculate each term according to the formula given; in this case, \( 3^i \cdot 1092 \).
• Add all the terms from the first to the last, which gives us the desired sum.

Understanding summation plays a crucial role in simplifying and solving many mathematical problems, especially in series and sequence calculations.

Geometric Progression

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a constant, known as the common ratio. In our exercise, the sequence is formed by powers of 3: \( 3^1, 3^2, 3^3, ..., 3^6 \). Here, the first term \( a \) is 3, and the common ratio \( r \) is also 3.

The magic of geometric progression is in its formula for finding the sum of its terms, which is especially useful when you're dealing with large numbers:

• The sum of the first \( n \) terms \( S_n \) of a geometric series is given by the formula:\[ S_n = a \cdot \frac{r^n - 1}{r - 1} \]
• In our solution, \( a = 3 \), \( r = 3 \), and \( n = 6 \).
• This helps us easily compute the sum without independently calculating each term in the series.

By applying this formula, we simplify the process of summation and solve the problem efficiently, maintaining the elegance of mathematical logic.

Arithmetic Calculations

Arithmetic calculations form the backbone of solving geometric series problems. They involve basic mathematical operations like addition, subtraction, multiplication, and division. Let's illustrate this with our exercise steps.

First, we factor out the constant 1092 from all terms in the summation, saving us from performing repetitive calculations. This leads to focusing solely on the geometric series: \( \sum_{i=1}^{6} 3^i \).

The calculations proceed as follows:

• Calculate \( 3^6 = 729 \).
• Use the geometric series sum formula: \[ S_6 = 3 \cdot \frac{729 - 1}{2} = 3 \cdot 364 = 1092 \]
• Finally, multiply the above sum by the constant: \( 1092 \times 1092 = 1192464 \).

These arithmetic steps ensure our solution is correct, reflecting how diligent calculations verify each stage of the process, particularly crucial when large numbers are involved.