Problem 54 - College Algebra

Question

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$a+a r+a r^{2}+\dots+a r^{n-1}$$

Step-by-Step Solution

Answer

The sum \(a+ar+ar^2+...+ar^{n-1}\) can be represented in summation notation as \(\sum_{i=0}^{n-1} ar^i\).

Step 1: Identifying the pattern

Observe the pattern in the given sum. We can see that each term of the sum after the first is obtained by multiplying the previous term by \(r\). This indicates that we are dealing with a geometric progression where \(a\) is the first term and \(r\) is the common ratio.

Step 2: Expressing specific terms using general formula

Expressing each term, the first term is \(a\), the second term is \(ar\), the third term is \(ar^2\), and so on. This shows that any term \((n-1)\) of this geometric progression can be expressed as \(ar^{i-1}\), where \(i\) is the term number.

Step 3: Writing the sum in summation notation

The sum can therefore be represented as a sum of terms from \(i=0\) to \(n-1\), every term being \(ar^i\). That is, \(\sum_{i=0}^{n-1} ar^i\).

Key Concepts

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the **common ratio**.
This type of sequence is common in many mathematical problems, particularly those involving exponential growth or decay. The structure is defined by its first term, denoted as \(a\), and the common ratio, denoted as \(r\).
In our original exercise, the formula \(a + ar + ar^2 + \dots + ar^{n-1}\) represents a geometric progression where each term is derived from its predecessor by multiplication by \(r\).

• The **first term** is \(a\)
• The **second term** becomes \(ar\)
• The **third term** is \(ar^2\), and so forth.

Recognizing this structure allows us to express such sums succinctly and efficiently using summation notation. Each of these terms contributes to the sequence formed by a geometric progression.

Sequence and Series

In mathematics, both sequences and series are essential concepts that help in understanding ordered lists of numbers and their sums.
A **sequence** is an ordered list of numbers following a particular rule, while a **series** is the sum of the terms of a sequence.
In a geometric progression, a sequence is expressed as \(a, ar, ar^2, \dots\) until the final term \(ar^{n-1}\).
When we sum the terms of this sequence, it results in a series, specifically called a **geometric series**. In our exercise, we deal with the series represented by the sum \(a + ar + ar^2 + \dots + ar^{n-1}\).
The way to effectively represent such a series is through summation notation, which allows us to condense the expression into a more manageable form.

Index of Summation

Summation notation introduces a systematic way to express the sum of a series using an index of summation.
This index typically starts at a specified lower limit, ascends by increments (usually by 1), and stops at the upper limit.
In the context of our exercise, the index of summation is represented by \(i\), starting from \(0\) and continuing up to \(n-1\).

• Here, \(i\) represents each term in the sequence, indicating the term number.
• The **expression** for each term is \(ar^i\) while importing into summation notation results in \(\sum_{i=0}^{n-1} ar^i\).

This concise notation greatly simplifies the process of summing sequences and provides a potent tool in solving a variety of mathematical problems, making it an invaluable part of mathematical notation.