Question

What is the difference between a geometric sum and a geometric series?

Short answer

Answer: A geometric series is an infinite sequence of terms with a constant ratio between them, while a geometric sum is the calculated value obtained by adding the terms of the geometric series up to a specified index n. A geometric sum is expressed as a formula, while a geometric series is an ordered list of terms.

Step-by-step solution

Define a geometric series

A geometric series is an infinite sequence of terms in which each term is the product of the previous term and a constant (common ratio, often denoted as "r"). In other words, a geometric series is formed when successive terms in the sequence are multiplied by the same constant. The general form of a geometric series can be written as: S_n = a_1 + a_2 + a_3 + ... + a_n = a_1 (1 + r + r^2 + ... + r^(n-1)) Here, S_n represents the sum of the first n terms in the series, a_1 is the first term, r is the common ratio between terms, and n is the total number of terms.

Define a geometric sum

A geometric sum is the result of adding up all the terms in a geometric series up to a specified index n. Since a geometric series is an ordered sequence of terms, a geometric sum is the value calculated by adding all those terms together up to the nth term. The formula for finding the sum of the first n terms of a geometric series (geometric sum) is: S_n = a_1 (1 - r^n) / (1 - r) Where S_n is the geometric sum of the first n terms, a_1 is the first term, r is the common ratio, and n is the total number of terms.

Highlight the differences

The main differences between a geometric sum and a geometric series are: 1. A geometric series is an infinite sequence of terms with a constant (common) ratio between them, while a geometric sum is the calculated value obtained by adding the terms of the geometric series up to a specified index n. 2. A geometric sum is expressed as a formula, which is a function of the first term (a_1), the common ratio (r), and the number of terms (n), whereas a geometric series is expressed as an ordered list of terms. In summary, a geometric series represents the sequence of terms with a constant ratio, while a geometric sum is the calculated value resulting from adding those terms together up to a certain point in the series.