To approach arithmetic and geometric sequences, let's break down both types and understand how to express them in two forms: recursively and explicitly. I'll also provide examples of situations where these sequences might be used, and show how to translate between the recursive and explicit forms.

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1. Arithmetic Sequence

In an arithmetic sequence, each term is obtained by adding a constant difference, $d$, to the previous term.

Recursive Form

The recursive formula for an arithmetic sequence with an initial term $a_1$ and common difference $d$ is: $a_n = a_{n-1} + d$ where $a_1$ is the first term.

Explicit Formula

The explicit formula (also called the closed form) for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1) \cdot d$

Example Situation: Savings

Imagine saving $100 each month, starting with an initial deposit of $500. The sequence representing your savings would be an arithmetic sequence with $a_1 = 500$ and $d = 100$.

• Recursive Formula: $S_n = S_{n-1} + 100$, with $S_1 = 500$
• Explicit Formula: $S_n = 500 + (n-1) \cdot 100 = 500 + 100n - 100 = 100n + 400$

This tells you that after any given month $n$, the total savings would be $100n + 400$ dollars.

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2. Geometric Sequence

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, $r$.

Recursive Form

The recursive formula for a geometric sequence with an initial term $a_1$ and common ratio $r$ is: $a_n = a_{n-1} \cdot r$ where $a_1$ is the first term.

Explicit Formula

The explicit formula for the $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$

Example Situation: Investment Growth

Suppose you invest $1000, and it grows by 5% each year. This is a geometric sequence with $a_1 = 1000$ and $r = 1.05$ (since 5% growth means multiplying by 1.05 each year).

• Recursive Formula: $I_n = I_{n-1} \cdot 1.05$, with $I_1 = 1000$
• Explicit Formula: $I_n = 1000 \cdot (1.05)^{n-1}$

This tells you that after $n$ years, your investment will be $1000 \cdot (1.05)^{n-1}$.

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Translating Between Recursive and Explicit Forms

For both sequences, translating from recursive to explicit form involves identifying either the common difference $d$ (for arithmetic) or the common ratio $r$ (for geometric), and using the formulas as shown.

1. Arithmetic Example: From $a_n = a_{n-1} + d$, use the explicit formula $a_n = a_1 + (n-1)d$.
2. Geometric Example: From $a_n = a_{n-1} \cdot r$, use the explicit formula $a_n = a_1 \cdot r^{n-1}$.

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Would you like more detailed examples or further help with a specific part?

Additional Questions

1. How can these sequence formulas be applied in compound interest calculations?
2. What are some real-life applications of recursive arithmetic sequences?
3. How can we derive the sum of an arithmetic sequence using its explicit form?
4. What methods exist for determining if a sequence is arithmetic or geometric?
5. How does changing the initial term affect the sequence and its formulas?

Tip

To quickly identify an arithmetic sequence, check if the difference between consecutive terms is constant; for geometric, verify if the ratio is consistent.