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## Algebra 1

### Course: Algebra 1>Unit 9

Lesson 6: General sequences

# Sequences: FAQ

## What is the difference between an arithmetic sequence and a geometric sequence?

In an arithmetic sequence, we add the same number to each term in order to get the next term in the sequence. In a geometric sequence, we multiply each term by the same number in order to get the next term.

## How do we construct arithmetic sequences?

Start with the first term of the sequence, which can be any number. Then, choose a common difference. This is the number we will add to each term in order to get the next term. For example, if we start with 5 and have a common difference of 3, our sequence will be 5, comma, 8, comma, 11, comma, 14, comma, 17, comma, 20, dots

## How do we construct geometric sequences?

Similar to arithmetic sequences, we start with the first term of the sequence. Then, we choose a common ratio. This is the number we will multiply each term by in order to get the next term. For example, if we start with 2 and have a common ratio of 3, our sequence will be 2, comma, 6, comma, 18, comma, 54, comma, 162, comma, 486, dots

## What is the difference between a recursive formula and an explicit formula for a sequence?

An explicit formula allows us to calculate the value of any term in the sequence by plugging in the term number, while a recursive formula defines each term in the sequence in relation to the terms that came before it.

## What is an example of a recursive formula?

A common example of a recursive formula is the formula for the Fibonacci sequence. The Fibonacci sequence starts with the two terms 0 and 1, and each subsequent term is found by adding the two most recent terms together:
0, comma, 1, comma, 1, comma, 2, comma, 3, comma, 5, comma, 8, comma, 13, comma, 21, comma, dots
So in this case, the recursive formula would be F, start subscript, n, end subscript, equals, F, start subscript, n, minus, 1, end subscript, plus, F, start subscript, n, minus, 2, end subscript.

## What is an example of an explicit formula?

An example of an explicit formula is the formula for the arithmetic sequence. For any arithmetic sequence, we can find the value of any term by using the formula a, start subscript, n, end subscript, equals, a, start subscript, 1, end subscript, plus, left parenthesis, n, minus, 1, right parenthesis, d, where a, start subscript, 1, end subscript is the first term in the sequence, d is the common difference, and n is the term number.

## Where can we use sequences?

Sequences are important in a variety of real-world applications. For example, financial analysts might use geometric sequences to calculate compound interest or to model the growth of an investment over time. Scientists might use arithmetic sequences to measure the rate at which something is changing. Sequences are also important in mathematics itself, as they can be used to understand patterns and relationships between numbers.