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# Series: FAQ

## What is a geometric series?

A geometric series is a sum of terms in which each term is found by multiplying the previous term by a constant number. We call this constant number the common ratio.

## What is an arithmetic series?

An arithmetic series is a sum of terms in which each term is found by adding a constant number to the previous term. We call this constant number the common difference.

## Where are geometric and arithmetic series used in the real world?

There are a lot of places! Geometric series are often used in finance and investment calculations, such as for compound interest. Arithmetic series can be used in a variety of ways, such as in scheduling or planning tasks. Both types of series are also used frequently in mathematics and physics for modeling and problem solving.

## What's the difference between a geometric and arithmetic series?

In a geometric series, you multiply each term by a constant number to get the next term. In an arithmetic series, you add a constant number to each term to get the next term.

## What is summation notation?

Summation notation is a shorthand way of writing out a sum of many terms. It uses the Greek letter $\mathrm{\Sigma }$ (capital sigma) to indicate the sum.

## What is the binomial theorem?

The binomial theorem allows us to expand expressions of the form $\left(a+b{\right)}^{n}$ in a systematic way.

## Want to join the conversation?

• This isn't a question, just a quick summary of this article. Arithmetic and geometric series are both important types of mathematical sequences. An arithmetic series is a sequence of numbers where each term is obtained by adding a constant value, called the common difference, to the previous term. In contrast, a geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a constant value, called the common ratio.

Summation notation, also known as sigma notation, is a shorthand way of writing the sum of a sequence without writing out all of the individual terms. The binomial theorem is a mathematical theorem that describes the expansion of powers of binomials. It gives a formula for the coefficients of each term in the expansion of (a+b)^n, where 'a' and 'b' are any two numbers, and 'n' is a non-negative integer.

Arithmetic and geometric series are used in various fields, such as finance and physics, to model and solve real-world problems involving growth rates, interest rates, and other related phenomena. Similarly, the binomial theorem has many applications in combinatorics, probability theory, and other areas of mathematics. It is also used in physics, engineering, and other sciences to model and solve problems involving binomial distributions and other related phenomena.
• Why are Series not included as units for mastery in Precalculus Course?
• how do you find the number of terms when provided the first and last term and common difference
• The 𝑛-th term of an arithmetic sequence can be written as
𝑎(𝑛) = 𝑎(1) + (𝑛 − 1)⋅𝑑, where 𝑎(1) is the first term of the sequence and 𝑑 is the common difference.

Solving for 𝑛, we get
𝑛 = 1 + (𝑎(𝑛) − 𝑎(1))∕𝑑

Example: In an arithmetic sequence the first term is 8, the last term is 35 and the common difference is 3.

The number of terms is then
𝑛 = 1 + (35 − 8)∕3 = 10