Question

using the geometric sequence 2, 8,32, 128, 512, 2048,...write a explicit formula for the geometric sequence, use your explicit formula to find the 100th term for the sequence

Answer 1

What is the numerical value of the 100th term in the sequence

Answer 2

Using the arithmetic sequence 1,4,7, 10, 13, 16, the explicit formula of the arithmetic sequence to find the 100th term in the sequence is a n = 1 + 3 ( 100 - 1)= 300, write a linear function to represent your arithmetic sequence show that it has the same structure as your explicit formula

Answer 3

To find the explicit formula for the geometric sequence given (2, 8, 32, 128, 512, 2048,...), we first need to identify the first term and the common ratio.

1. The **first term** \( a \) is \( 2 \).
2. To find the **common ratio** \( r \), we divide the second term by the first term:
\[
r = \frac{8}{2} = 4
\]

Now we can write the explicit formula for the \( n \)-th term of a geometric sequence, which is given by:
\[
a_n = a \cdot r^{n-1}
\]
Substituting the values we found:
\[
a_n = 2 \cdot 4^{n-1}
\]

Now we will use this formula to find the 100th term, \( a_{100} \):
\[
a_{100} = 2 \cdot 4^{100-1} = 2 \cdot 4^{99}
\]

We can simplify this further. Since \( 4 = 2^2 \), we have:
\[
a_{100} = 2 \cdot (2^2)^{99} = 2 \cdot 2^{198} = 2^{1 + 198} = 2^{199}
\]

Now, we can calculate \( 2^{199} \). Depending on the context, you might need to give the result as a power or compute its numerical value.

The final answer for the 100th term of the sequence is:
\[
a_{100} = 2^{199}
\]

If you want the numerical value, \( 2^{199} \) can be calculated with a calculator or programming tool. It is a very large number, approximately \( 6.338 \times 10^{59} \).

Answer 4

To calculate the numerical value of the 100th term in the sequence \( a_{100} = 2^{199} \):

We can compute \( 2^{199} \). Using a calculator or programming language capable of handling large integers:

\[
2^{199} = 633825300114114700748351602688
\]

Thus, the numerical value of the 100th term in the sequence is:

\[
\boxed{633825300114114700748351602688}
\]

Answer 5

To develop a linear function that represents the arithmetic sequence \(1, 4, 7, 10, 13, 16, \ldots\), we first identify the elements of the sequence:

1. The **first term** \( a \) is \( 1 \).
2. The **common difference** \( d \) is \( 3 \) (as \( 4 - 1 = 3 \), \( 7 - 4 = 3 \), etc.).

### Explicit Formula
The explicit formula for the \( n \)-th term \( a_n \) of an arithmetic sequence is given by:
\[
a_n = a + (n - 1) \cdot d
\]
Plugging in the values:
\[
a_n = 1 + (n - 1) \cdot 3
\]
This simplifies to:
\[
a_n = 1 + 3(n - 1) = 1 + 3n - 3 = 3n - 2
\]

So, the explicit formula for the \( n \)-th term is:
\[
a_n = 3n - 2
\]

### Linear Function Representation
In terms of a linear function, we can express the sequence as follows:
Let \( f(n) \) be a function that gives us the \( n \)-th term of the sequence:
\[
f(n) = 3n - 2
\]

This function \( f(n) \) is indeed linear because it can be represented in the standard linear function form \( f(x) = mx + b \), where:
- \( m = 3 \) (the slope, which represents the common difference)
- \( b = -2 \) (the y-intercept)

### Validation of Structure
To show that this linear function has the same structure as our explicit formula:
1. The explicit formula gives \( a_n = 3n - 2 \).
2. The linear function \( f(n) = 3n - 2 \) also yields the same expression.

Thus, both representations are equivalent and describe the same arithmetic sequence.

### Finding the 100th Term
Using either formula to find the 100th term:
1. From the explicit formula:
\[
a_{100} = 3(100) - 2 = 300 - 2 = 298
\]
2. From the linear function:
\[
f(100) = 3(100) - 2 = 300 - 2 = 298
\]

In both cases, the 100th term in the sequence is:
\[
\boxed{298}
\]