D. Create your own geometric sequence. List at least six terms of your sequence.

Question

Explain what makes it geometric.
E. Write a recursive formula for your geometric sequence.
F. Write an explicit formula for your geometric sequence. Use your explicit
formula to find the 100th term of your sequence.

Answer 1

D. Creating a Geometric Sequence

Let’s create a geometric sequence where the first term is \( a = 3 \) and the common ratio \( r = 2 \).

The sequence can be listed as follows:

1. First term: \( a_1 = 3 \)
2. Second term: \( a_2 = a_1 \cdot r = 3 \cdot 2 = 6 \)
3. Third term: \( a_3 = a_2 \cdot r = 6 \cdot 2 = 12 \)
4. Fourth term: \( a_4 = a_3 \cdot r = 12 \cdot 2 = 24 \)
5. Fifth term: \( a_5 = a_4 \cdot r = 24 \cdot 2 = 48 \)
6. Sixth term: \( a_6 = a_5 \cdot r = 48 \cdot 2 = 96 \)

So, the sequence is: 3, 6, 12, 24, 48, 96

This sequence is geometric because each term after the first is found by multiplying the previous term by a constant (in this case, the common ratio \( r = 2 \)).

E. Recursive Formula

The recursive formula for a geometric sequence can be written as:

\[ a_n = a_{n-1} \cdot r \]

where:

• \( a_1 = 3 \) (the first term),
• \( r = 2 \) (the common ratio).

Thus, the recursive formula for our sequence is:

\[ \begin{align*} a_1 &= 3 \ a_n &= a_{n-1} \cdot 2 \quad \text{for } n > 1 \end{align*} \]

F. Explicit Formula

The explicit formula for a geometric sequence can be written as:

\[ a_n = a \cdot r^{n-1} \]

For our sequence, this gives us:

\[ a_n = 3 \cdot 2^{n-1} \]

To find the 100th term (\( a_{100} \)), we substitute \( n = 100 \) into the explicit formula:

\[ a_{100} = 3 \cdot 2^{100-1} = 3 \cdot 2^{99} \]

Calculating this value:

\[ 2^{99} \approx 6.338 \times 10^{29} \quad (\text{as an approximation}) \] Thus,

\[ a_{100} \approx 3 \cdot (6.338 \times 10^{29}) \approx 1.9014 \times 10^{30} \]

So, the 100th term of the sequence is approximately \( 1.9014 \times 10^{30} \).