Question
Write a recursive formula for your geometric sequence. 3, 6, 12, 24, 48, 96
Answers
Answered by
GPT-4o mini
To define a recursive formula for the given geometric sequence \(3, 6, 12, 24, 48, 96\), we first identify the pattern in the sequence.
This sequence has a common ratio of \( r = 2 \) (each term is twice the previous term).
To express this sequence recursively, we need to state the first term and the formula that relates each term to the one before it.
Let \( a_n \) represent the \( n \)-th term in the sequence.
1. The first term \( a_1 \) is \( 3 \).
2. For \( n \geq 1 \), each subsequent term can be defined as:
\[
a_n = 2 \cdot a_{n-1}
\]
Putting it all together, the recursive formula for the sequence is:
\[
\begin{align*}
a_1 & = 3 \\
a_n & = 2 \cdot a_{n-1} \quad \text{for } n \geq 2
\end{align*}
\]
This sequence has a common ratio of \( r = 2 \) (each term is twice the previous term).
To express this sequence recursively, we need to state the first term and the formula that relates each term to the one before it.
Let \( a_n \) represent the \( n \)-th term in the sequence.
1. The first term \( a_1 \) is \( 3 \).
2. For \( n \geq 1 \), each subsequent term can be defined as:
\[
a_n = 2 \cdot a_{n-1}
\]
Putting it all together, the recursive formula for the sequence is:
\[
\begin{align*}
a_1 & = 3 \\
a_n & = 2 \cdot a_{n-1} \quad \text{for } n \geq 2
\end{align*}
\]
Answer
without using these symbols: \ * } { /
Answered by
GPT-4o mini
Sure! Here's the recursive formula for the geometric sequence without using the specified symbols:
1. The first term, which is the value at position one, is three.
2. For any term at position n greater than or equal to two, that term is equal to two times the value of the term at the previous position.
Putting this together, we can describe it as follows:
- The first term (a one) is equal to three.
- For any term (a n) where n is two or greater, it is equal to two times the previous term (a n minus one).
This gives you the recursive definition of the sequence.
1. The first term, which is the value at position one, is three.
2. For any term at position n greater than or equal to two, that term is equal to two times the value of the term at the previous position.
Putting this together, we can describe it as follows:
- The first term (a one) is equal to three.
- For any term (a n) where n is two or greater, it is equal to two times the previous term (a n minus one).
This gives you the recursive definition of the sequence.
Answer
do it recursive now please
Answered by
GPT-4o mini
Certainly! Here’s the recursive formula for the given geometric sequence presented in plain language without special symbols:
- The first term, which we will call "a one," is equal to three.
- For any term "a n" where "n" is two or greater, the value of "a n" is equal to two times the value of the previous term, which is "a n minus one."
In summary:
- a one is three.
- a n is two times a n minus one for n equal to two or greater.
- The first term, which we will call "a one," is equal to three.
- For any term "a n" where "n" is two or greater, the value of "a n" is equal to two times the value of the previous term, which is "a n minus one."
In summary:
- a one is three.
- a n is two times a n minus one for n equal to two or greater.
Answer
what does the look like
Answered by
GPT-4o mini
Here’s how the recursive formula for the geometric sequence looks when written out clearly:
1. a₁ = 3
(This means the first term is 3.)
2. aₙ = 2 × aₙ₋₁ for n ≥ 2
(This means that each term aₙ is two times the previous term aₙ₋₁ for all terms starting from the second term onward.)
So, if you list it out in a structured way:
- **First Term:**
- a₁ = 3
- **Recursive Formula:**
- aₙ = 2 × aₙ₋₁, for n ≥ 2
This captures the essence of the recursive relationship in the sequence you provided.
1. a₁ = 3
(This means the first term is 3.)
2. aₙ = 2 × aₙ₋₁ for n ≥ 2
(This means that each term aₙ is two times the previous term aₙ₋₁ for all terms starting from the second term onward.)
So, if you list it out in a structured way:
- **First Term:**
- a₁ = 3
- **Recursive Formula:**
- aₙ = 2 × aₙ₋₁, for n ≥ 2
This captures the essence of the recursive relationship in the sequence you provided.
Answer
what would this like for finding the 100 term
Answered by
GPT-4o mini
To find the 100th term of the given geometric sequence using the recursive formula requires knowing the first term and how the terms relate to each other.
Given the recursive formula:
1. a₁ = 3
2. aₙ = 2 × aₙ₋₁ for n ≥ 2
You can manually apply this formula to find each term up to the 100th term, but that would be tedious. Instead, there is a more efficient way to find the 100th term directly.
Since this is a geometric sequence with a common ratio, a more straightforward formula exists. The general term for a geometric sequence can be expressed as:
**aₙ = a₁ × r^(n - 1)**
Where:
- a₁ is the first term (3 in this case),
- r is the common ratio (2),
- n is the term number (100 for this case).
So, for the 100th term:
a₁ = 3
r = 2
n = 100
The formula becomes:
**a₁ = 3 × 2^(100 - 1)**
**a₁ = 3 × 2^(99)**
Now you can calculate \( a_{100} \):
\( a_{100} = 3 × 2^{99} \)
This expression gives you the value of the 100th term without needing to calculate all the preceding terms individually.
Given the recursive formula:
1. a₁ = 3
2. aₙ = 2 × aₙ₋₁ for n ≥ 2
You can manually apply this formula to find each term up to the 100th term, but that would be tedious. Instead, there is a more efficient way to find the 100th term directly.
Since this is a geometric sequence with a common ratio, a more straightforward formula exists. The general term for a geometric sequence can be expressed as:
**aₙ = a₁ × r^(n - 1)**
Where:
- a₁ is the first term (3 in this case),
- r is the common ratio (2),
- n is the term number (100 for this case).
So, for the 100th term:
a₁ = 3
r = 2
n = 100
The formula becomes:
**a₁ = 3 × 2^(100 - 1)**
**a₁ = 3 × 2^(99)**
Now you can calculate \( a_{100} \):
\( a_{100} = 3 × 2^{99} \)
This expression gives you the value of the 100th term without needing to calculate all the preceding terms individually.