Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ \frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{6561} $$

Step-by-Step Solution

Answer

The given sequence is a geometric sequence with the first term 'a' being \(\frac{2}{3}\) and the common ratio 'r' being \(\frac{1}{3}\).

Step 1: Identify The Pattern

Looking at the progression \( \frac{2}{3}, \frac{2}{9}, \frac{2}{27},... \), it can be noted that each term is being divided by 3 to obtain the next term. Hence, the sequence is a geometric progression.

Step 2: Identify 'a' And 'r'

In a geometric sequence, 'a' represents the first term in the sequence and 'r' the ratio between two successive terms. Here the first term \(a = \frac{2}{3} \) and the ratio \( r = \frac{\frac{2}{9}}{\frac{2}{3}} = \frac{1}{3} \).

Key Concepts

geometric progression

A geometric progression, also known as a geometric sequence, is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant called the "common ratio". This kind of sequence is distinct from arithmetic sequences, where the difference between terms is constant. In a geometric progression:

• Each successive term is derived from the previous one by multiplication.
• This multiplication factor is consistent and is referred to as the "common ratio".

The general formula for the terms of a geometric sequence can be written as \( a, ar, ar^2, ar^3, \ldots \), where \( a \) is the first term and \( r \) is the common ratio. Recognizing a geometric progression is key, as it allows us to analyze the sequence's properties and calculate further terms or even sums.

common ratio

The common ratio, denoted as \( r \), is a crucial part of understanding geometric sequences. It is the constant factor that you multiply by to get from one term to the next in the sequence. You can find the common ratio by dividing any term in the sequence by the preceding term. For example, in the sequence \( \frac{2}{3}, \frac{2}{9}, \frac{2}{27}, \ldots \), the common ratio \( r \) is calculated as follows:- Take any consecutive terms, such as \( \frac{2}{9} \) and \( \frac{2}{3} \).- Divide them: \( \frac{2}{9} \div \frac{2}{3} = \frac{1}{3} \).This demonstrates that \( r = \frac{1}{3} \), and each term is essentially a third of the previous one. Understanding the common ratio helps in predicting the sequence's behavior and in calculating sums efficiently.

first term

The first term of a geometric sequence, often represented by \( a \), serves as the starting point of the sequence. It is the initial value from which all other terms are derived by successive multiplication with the common ratio \( r \).In the context of the example sequence \( \frac{2}{3}, \frac{2}{9}, \frac{2}{27}, \ldots \):- The first term is \( a = \frac{2}{3} \).Identifying the first term is fundamental since it forms the base value upon which the geometric progression is built. And when we add or sum such a sequence, this term directly defines the baseline for calculations using the formulas of the geometric series.

sum of series

The sum of a geometric series refers to adding up all the terms of a geometric sequence. The formula for calculating the sum depends on whether the series is finite or infinite.For a finite geometric series, the sum \( S_n \) of the first \( n \) terms is given by:\[S_n = a \frac{1-r^n}{1-r}\]Here:- \( a \) is the first term,- \( r \) is the common ratio,- \( n \) is the number of terms.For an infinite geometric series, where \( |r| < 1 \), the sum \( S \) can be found using:\[S = \frac{a}{1-r}\]This formula only applies if the common ratio's absolute value is less than 1, ensuring the series converges. Understanding the sum formula allows computation of the total of a series or helps analyze convergence in infinite sequences.