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### 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 $5,8,11,14,17,20\dots$

## 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 $2,6,18,54,162,486\dots$

## 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:
$0,1,1,2,3,5,8,13,21,\dots$
So in this case, the recursive formula would be ${F}_{n}={F}_{n-1}+{F}_{n-2}$.

## 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 ${a}_{n}={a}_{1}+\left(n-1\right)d$, where ${a}_{1}$ 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.

## Want to join the conversation?

• Wouldn't it make more sense to call arithmetic sequences "linear sequences"? And maybe geometric sequences are "exponential sequences"? Or is that wrong?
• Correct. Linear sequences mean adding / subtracting the same value from the previous term to get the current term, which is the definition of arithmetic sequence.
Exponential sequences mean multiplying or dividing the same value from the previous term to get the current term, also the definition of geometric sequence.
However if you are asking about the context in this article, the way they assigned Recursive and Explicit to the formulas is correct.
• The notes above say, "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."

If this is the case, then why in one of the examples does the solution show that the first term in the sequence is 8, the domains in the multiple choice claimed that we can come to that by plugging in 2, 3, and 4. Shouldn't it only be 1 that works out to the first term of the sequence?
• I think it depends on the question structure.
• What is a recursive and explicit?please help,and thank you very much 3
• Recursive formula means you need to compute all required previous terms in the sequence for the formula in order to find the next term.
In the example, Fn = Fn-1 + Fn-2. You simply cannot find (Actually there's a formula but not necessary to mention it now) any term like the F10 term directly. You need to find F9 and F8, which leads to finding F7 and F6 etc.
Explicit formula enables you to just substitute known data and you can get that term in the sequence directly without calculating the terms before it.
• can you explain more ?
• what would you like to know?
• How do you write a sequence function?
• To write an explicit function for the sequence {40, 60, 80, ...}, the sequence increases by 20 each time meaning 'a' will be 20, but 1*20 doesn't equal the starting term so by subtracting 'a' from the starting term you get 'c'. By putting 'a' and 'c' in a format something along the lines of f(n)=an+c, you can write the explicit equation for it.

To write a recursive function, it's similar except now there is another variable. To put it simply, most of the time 'b' is just 1 because most of the time recursive functions define the current term using the previous term. To write a recursive fuction just write something like f(n)=a(n-b)+c.

In non-arithmetic instances of sequences there will be powers and other stuff but I don't know how to explain those without writing a massive chunk of text that will probably be very unclear and almost incomprehensible.
• Can someone explain the domain lesson to me? please HELP!!