Problem 93 - Algebra and Trigonometry

Question

What is the difference between a geometric sequence and an infinite geometric series?

Step-by-Step Solution

Answer

A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number while an infinite geometric series is the sum of an infinite geometric sequence. An infinite geometric series will converge to a finite number if the absolute value of the common ratio is less than one.

Step 1: Define the Geometric Sequences

A geometric sequence is an ordered list 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. For example, \(2, 6, 18, 54, ...\). The general form of a geometric sequence is \(a, ar, ar^2, ar^3, ...\) where 'a' is the first term (non-zero), 'r' is the common ratio (non-zero).

Step 2: Define the Infinite Geometric Series

An infinite geometric series is the sum of the terms of an infinite geometric sequence. It is expressed as \(a + ar + ar^2 + ar^3 + ...\). An infinite geometric series has a sum if the common ratio 'r' is between -1 and 1. The sum (S) of an infinite geometric series can be calculated by the formula \(S = a / (1 - r)\).

Step 3: Differentiate between Geometric Sequences and Infinite Geometric Series

Geometric sequence refers to the list of numbers in order where each term is obtained by multiplying the previous one by a constant ratio, whereas an infinite geometric series is the sum of the terms of this sequence, with the condition that the absolute value of the common ratio should be less than 1. In other words, a geometric sequence focuses on the individual terms while an infinite geometric series focuses on the sum of these terms.