Create your own arithmetic sequence. List at least six terms of your sequence. Explain what makes it arithmetic. State the fifth term of your arithmetic sequence using subscript notation. Write an explicit formula for your arithmetic sequence. Use your explicit formula to find the 100th term of your sequence. Create your own geometric sequence. List at least six terms of your sequence. Explain what makes it geometric. Write a recursive formula for your geometric sequence. Write an explicit formula for your geometric sequence. Use your explicit formula to find the 100th term of your sequence. Question 2 Graph your arithmetic sequence on a coordinate grid (a sheet of graph paper). Write a linear function to represent your arithmetic sequence. Show that it has the same structure as your explicit formula from Question 1C. Graph your geometric sequence on a coordinate grid (a sheet of graph paper). Write an exponential function to represent your geometric sequence. Show that it has the same structure as your explicit formula from Question 1F.

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GPT-4o mini · Answer

### Arithmetic Sequence **Sequence:** 3, 7, 11, 15, 19, 23 **Explanation:** An arithmetic sequence is defined by the property that the difference between consecutive terms is constant. In this case, the common difference (d) is 4 (7 - 3 = 4, 11 - 7 = 4, etc.). **Fifth Term in Subscript Notation:** The fifth term can be expressed as $ a_5 = 19 $. **Explicit Formula:** The explicit formula for an arithmetic sequence can be expressed as: $$ a_n = a_1 + (n - 1) \cdot d $$ Where $ a_1 $ is the first term, $ n $ is the term number, and $ d $ is the common difference. For our sequence: $$ a_n = 3 + (n - 1) \cdot 4 $$ Simplifying this gives: $$ a_n = 4n - 1 $$ **Finding the 100th Term:** Using the explicit formula: $$ a_{100} = 4(100) - 1 = 400 - 1 = 399 $$ --- ### Geometric Sequence **Sequence:** 2, 6, 18, 54, 162, 486 **Explanation:** A geometric sequence is characterized by having a constant ratio between consecutive terms. In this case, the common ratio (r) is 3 (6 / 2 = 3, 18 / 6 = 3, etc.). **Recursive Formula:** The recursive formula for a geometric sequence can be expressed as: $$ a_n = r \cdot a_{n-1} $$ For our sequence: $$ a_n = 3 \cdot a_{n-1} $$ with the initial condition $ a_1 = 2 $. **Explicit Formula:** The explicit formula for a geometric sequence can be written as: $$ a_n = a_1 \cdot r^{(n - 1)} $$ For our sequence: $$ a_n = 2 \cdot 3^{(n - 1)} $$ **Finding the 100th Term:** Using the explicit formula: $$ a_{100} = 2 \cdot 3^{(100 - 1)} = 2 \cdot 3^{99} $$ --- ### Question 2 **Graphing the Arithmetic Sequence:** To graph the arithmetic sequence $ 3, 7, 11, 15, 19, 23 $, you can plot the points (1, 3), (2, 7), (3, 11), (4, 15), (5, 19), and (6, 23) on a coordinate grid. **Linear Function:** The linear function representing the arithmetic sequence can be written as: $$ f(x) = 4x - 1 $$ This function has the same structure as the explicit formula $ a_n = 4n - 1 $. --- **Graphing the Geometric Sequence:** To graph the geometric sequence $ 2, 6, 18, 54, 162, 486 $, you can plot the points (1, 2), (2, 6), (3, 18), (4, 54), (5, 162), and (6, 486) on a coordinate grid. **Exponential Function:** The exponential function representing the geometric sequence can be written as: $$ g(x) = 2 \cdot 3^{(x - 1)} $$ This function has the same structure as the explicit formula $ a_n = 2 \cdot 3^{(n - 1)} $. These graphs will visually demonstrate the linear growth of the arithmetic sequence and the exponential growth of the geometric sequence.