Problem 68 - Algebra and Trigonometry

Question

Express each sum using summation notation. \(a+a r+a r^{2}+\cdots+a r^{n-1}\)

Step-by-Step Solution

Answer

\( \sum_{k=1}^{n} a r^{k-1} \)

Step 1: Identify the Series

Recognize the given series: \[ a + ar + ar^2 + \cdots + ar^{n-1}\]. This is a geometric series.

Step 2: Identify the First Term and Common Ratio

The first term \(a_1\) is \(a\) and the common ratio \(r\) is \(r\).

Step 3: Determine the General Term

The general term of a geometric series is \(a r^{k-1}\), where \(k\) is the index of summation starting from 1.

Step 4: Set Up Summation Notation

Convert the series to summation notation: \[ \sum_{k=1}^{n} a r^{k-1} \].

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. This concept is crucial in various fields of study, including mathematics, physics, and finance.
For example, in the series \[ a + ar + ar^2 + \cdots + ar^{n-1} \], each term is obtained by multiplying the previous term by the common ratio \( r \).
Recognizing a geometric series helps in understanding its properties and finding solutions efficiently.

First Term

In any geometric series, the first term is a vital part because it sets the starting point for the sequence. The first term is denoted by \( a \).
For instance, in the series \[ a + ar + ar^2 + \cdots + ar^{n-1} \], the first term is \( a \).
This term is crucial because the whole series is built from it. Changing the first term changes all subsequent terms even if the common ratio remains the same.
Practically, it's the value you begin with before you start multiplying by the common ratio.

Common Ratio

The common ratio is the factor by which we multiply each term to get the next term in a geometric series. It is represented by \( r \).
For example, in the series \[ a + ar + ar^2 + \cdots + ar^{n-1} \], \( r \) is the common ratio.
This ratio is essential because it determines the rate at which the terms grow or shrink.
If \( r \) is greater than 1, the series will grow; if \( r \) is between 0 and 1, the series will get smaller. If \( r \) is negative, the terms will alternate between positive and negative.

General Term

The general term in a geometric series allows us to find any term in the sequence without listing all the preceding terms. It is given by the formula \[ a r^{k-1} \], where \( a \) is the first term, \( r \) is the common ratio, and \( k \) is the position of the term in the sequence.
For instance, in the series \[ a + ar + ar^2 + \cdots + ar^{n-1} \], the general term can be expressed as \[ ar^{k-1} \].
This formula is helpful because you can calculate any term directly, which is faster and more efficient than listing all terms up to the desired one.