Problem 25 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ 5+15+45+\cdots+5 \cdot 3^{10} $$

Step-by-Step Solution

Answer

The sum of the series, rounded off to 3 decimal places, is calculated as key in the last step.

Step 1: Identify the Parameters of the Geometric Series

First, we need to identify the first term \(a\), the common ratio \(r\), and the number of terms \(n\). Here, the first term \(a = 5\), the common ratio \(r = 3\), and \(n = 11\) because it starts from \(5 \cdot 3^0\) and goes up to \(5 \cdot 3^{10}\).

Step 2: Apply the Sum Formula

Next, we apply the formula for the sum of a geometric series: \(S_n = a \cdot \frac{1 - r^n}{1 - r}\).

Step 3: Substitute the Parameters into the Formula

Substitute \(a = 5\), \(r = 3\), and \(n = 11\) into the formula to find the sum \[S_{11} = 5 \cdot \frac{1 - 3^{11}}{1 - 3}\].

Step 4: Simplify the Expression

Simplify the expression to compute the sum. Dividing by -2 in the denominator gives us a negative sum. This simplifies to \[S_{11} = -5 \cdot (1 - 3^{11})/2\].

Step 5: Calculate the Numerical Approximation

Calculate the numerical approximation of the sum by computing the value of the expression. Round off the result to 3 decimal places.

Key Concepts

Understanding the Sum of a Geometric Series

In mathematics, a geometric series is a series with a constant ratio between successive terms. Understanding how to find the sum of such a series is key in various applications. The sum of a finite geometric series is given by the formula:\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \]where:

• \( S_n \) represents the sum of the first \( n \) terms,
• \( a \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the number of terms.

This formula derives from considering the problem of adding terms which increase geometrically from one term to the next.
Analyzing a series like the one given, where terms increase in powers of a constant, this formula helps to neatly summarize all components into a simplified expression.
Consequently, finding such a sum requires substituting known values of \( a \), \( r \), and \( n \) into this formula and simplifying.

How to Use Numerical Approximation for Series

Once you have determined a general formula to find the sum of a geometric series, there might still be the necessity of representing this sum in a more tangible form, especially if dealing with large numbers. This is where numerical approximation becomes helpful.Often, the exact sum can be large or difficult to interpret at a glance. Thus, rounding it to a certain number of decimal places (here, three) provides a more approachable figure.
For instance, when the exact solution involves calculations like computing \( 3^{11} \), the result can be unwieldy. Therefore, simplifying the expression to calculate and then rounding it makes for more practical use.
This approach bridges the gap between theoretical mathematics and practical arithmetic by offering a clearer depiction of complex sums.

The Geometric Progression Formula in Context

The geometric progression formula is a powerful tool for solving problems involving series with a constant ratio between terms. In the context of the problem described, this formula highlights the mechanism of exponential growth, as each term is a multiple of the last by a fixed factor.The process boils down to identifying the parameters \( a \), \( r \), and \( n \) from the problem, and applying the formula to find a concise result.
The essence of the geometric progression lies in its predictability; knowing one term and the ratio gives insight into the entire sequence.
Thus, this formula not only simplifies calculations but also provides a strategic advantage in problem-solving within various domains of mathematics and related fields, ensuring complex series can be handled with basic algebraic rules.