Question

In Exercises 47–52, find the sum. $$ \sum_{i=1}^{10} 4\left(\frac{3}{4}\right)^{i-1} $$

Short answer

The sum of the series \( \sum_{i=1}^{10} 4(\frac{3}{4})^{i-1} \) is 16.

Step-by-step solution

Identify the components of the series

First, let's identify the components in the given sum based on the formula for a geometric series. In this case, the first term \(a = 4\), the common ratio \(r = \frac{3}{4}\), and there are \(n = 10\) terms in total.

Substitute into the formula

Now, let's substitute these values into the formula for the sum of a geometric series. This leads to \(S= 4*(1 - (\frac{3}{4})^{10}) / (1 - \frac{3}{4})\).

Calculate the sum

Simplify the above expression to find the sum. The sum S expresses to \(S = 16\).

Key concepts

Sum of a Series

In mathematics, the sum of a series refers to the total value obtained by adding all terms in a sequence. When discussing a geometric series, this sequence is composed of terms where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The specific formula for the sum of a finite geometric series with first term \(a\), common ratio \(r\), and \(n\) number of terms is:

• \[ S_n = a \frac{1 - r^n}{1 - r} \]

This formula is crucial as it allows us to quickly calculate the sum without manually adding each term, especially as the number of terms grows large. This facilitates easier computations in mathematical problems, such as the one given in the exercise, where the rapid summation is necessary.

Common Ratio

The common ratio in a geometric series is a key component that determines the progression of the series. In simpler terms, it is the constant factor by which we multiply one term of the series to obtain the subsequent term. For the series in the given exercise, the common ratio \(r\) is \(\frac{3}{4}\).
Understanding the role of the common ratio can:

• Help identify how quickly the terms of a series are increasing or decreasing.
• Determine if the series converges, especially in infinite series scenarios where \(|r| < 1\).
• Significantly simplify the calculation of the series sum using the geometric sum formula.

The common ratio is a fundamental concept in Algebra 2, where identifying it leads to the effective application of various formulas and results.

Algebra 2

Algebra 2 is a branch of mathematics that deals with complex equations, functions, and analyzing relationships between variables. It extends the concepts learned in introductory algebra to a more advanced level, including the study of sequences and series like geometric and arithmetic series. In Algebra 2, students engage with concepts such as:

• Linear, quadratic, and exponential functions.
• Polynomials and rational expressions.
• Deep exploration of geometric and arithmetic series.

In the context of this exercise, Algebra 2 introduces students to solving and manipulating series using algebraic methods, enabling them to understand the behavior and sum of geometric series effectively.

Mathematical Formulas

Mathematical formulas are the language used by mathematicians to express relationships between quantities succinctly. They are vital for solving problems efficiently and are a critical tool in both pure and applied mathematics. The given problem employs the formula for the sum of a geometric series:

• \[ S_n = a \frac{1 - r^n}{1 - r} \]

Utilizing this formula makes it straightforward to calculate the final sum without unnecessary computation. In the given example, substituting \(a = 4\), \(r = \frac{3}{4}\), and \(n = 10\) into the formula renders a quick solution.
Understanding and memorizing key formulas allow students to approach algebra problems systematically and confidently, reducing potential errors and increasing efficiency in calculations.