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Chapter 15: Problem 1

What is the difference between an arithmetic and a geometric series?

Short Answer

Expert verified
In an arithmetic series, terms are generated by adding a constant difference to each term, with the general form \(a, a + d, a + 2d, a + 3d, ..., a + nd\). In a geometric series, terms are generated by multiplying each term by a constant ratio, with the general form \(a, ar, ar^2, ar^3, ..., ar^n\).

Step by step solution

01

Arithemetic Series Definition

An arithmetic series is a sequence of numbers formed by adding a constant difference to each term. In other words, in an arithmetic series, the difference between any two consecutive terms is always the same. The general form of an arithmetic series is given by \(a, a + d, a + 2d, a + 3d, ..., a + nd\), where 'a' is the first term, 'd' is the constant difference between terms, and 'n' is the number of terms.
02

Geometric Series Definition

A geometric series is a sequence of numbers formed by multiplying each term by a constant factor to obtain the next term. In other words, in a geometric series, the ratio of any two consecutive terms is always the same. The general form of a geometric series is given by \(a, ar, ar^2, ar^3, ..., ar^n\), where 'a' is the first term, 'r' is the constant ratio between terms, and 'n' is the number of terms.
03

Arithemetic Series Example

An example of an arithmetic series could be the sequence of numbers: \(2, 5, 8, 11, 14\). Here, the first term 'a' is 2, the common difference 'd' is 3, and the number of terms 'n' is 5. Each successive term in the sequence is obtained by adding the common difference 3 to the previous term.
04

Geometric Series Example

An example of a geometric series could be the sequence of numbers: \(2, 6, 18, 54, 162\). Here, the first term 'a' is 2, the common ratio 'r' is 3, and the number of terms 'n' is 5. Each successive term in the sequence is obtained by multiplying the previous term by the common ratio 3.
05

Summary

In summary, the main difference between an arithmetic and a geometric series lies in the method used to generate the terms in the series. In an arithmetic series, terms are generated by adding a constant difference to each term, whereas in a geometric series, terms are generated by multiplying each term by a constant ratio.

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