Step 1: Create your own arithmetic sequence. List at least six terms of your sequence.

Question

Explain what makes it arithmetic.

Step 2: State the fifth term of your arithmetic sequence using subscript notation.

Step 3: Write an explicit formula for your arithmetic sequence. Use your explicit
formula to find the 100th term of your sequence.

Step 4: Create your own geometric sequence. List at least six terms of your sequence.
Explain what makes it geometric.

Step 5: Write a recursive formula for your geometric sequence.

Step 6: Write an explicit formula for your geometric sequence. Use your explicit
formula to find the 100th term of your sequence.

Answer each step seperately
Simplify your answer to be as short as possible

Answer 1

Step 1:

Arithmetic Sequence: 2, 5, 8, 11, 14, 17

Explanation: This sequence is arithmetic because each term is obtained by adding a constant difference of 3 (the common difference) to the previous term.

---

Step 2:

Fifth Term: \( a_5 = 14 \)

---

Step 3:

Explicit Formula: The explicit formula for the nth term is given by \( a_n = a_1 + (n - 1)d \), where \( a_1 = 2 \) and \( d = 3 \). Thus, \( a_n = 2 + (n - 1) \cdot 3 = 3n - 1 \).

Finding the 100th Term: \[ a_{100} = 3(100) - 1 = 300 - 1 = 299 \]

---

Step 4:

Geometric Sequence: 3, 6, 12, 24, 48, 96

Explanation: This sequence is geometric because each term is obtained by multiplying the previous term by a constant ratio of 2 (the common ratio).

---

Step 5:

Recursive Formula:

• \( a_1 = 3 \)
• \( a_n = a_{n-1} \cdot 2 \) for \( n > 1 \)

---

Step 6:

Explicit Formula: The explicit formula for the nth term is given by \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 = 3 \) and \( r = 2 \). Thus, \( a_n = 3 \cdot 2^{(n-1)} \).

Finding the 100th Term: \[ a_{100} = 3 \cdot 2^{99} \]

This value is extremely large, but it evaluates to: \[ a_{100} = 3 \cdot 633825300114114700748351602688 = 1901475900341964102245054808064 \]

You can express \( a_{100} \) as approximately \( 1.9014759 \times 10^{36} \).