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

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 2

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

Answer 3

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 4

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 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: \[ \boxed{298} \]