Solved Problems: Writing Explicit Formulas from Sequence Patterns

📚 Understanding Explicit Formulas for Sequences

An explicit formula allows you to directly calculate any term in a sequence without knowing the preceding terms. It's a powerful tool for understanding and working with sequences. Think of it as a 'plug-and-chug' formula where you input the term number (usually denoted by 'n') and get the value of that term.

📜 A Brief History

The concept of sequences and series has ancient roots, dating back to early civilizations like the Babylonians and Greeks, who studied arithmetic and geometric progressions. The formalization of explicit formulas came later with the development of algebraic notation and calculus.

🔑 Key Principles for Writing Explicit Formulas

• 🔍 Identify the Pattern: Look for a common difference (arithmetic sequence), a common ratio (geometric sequence), or a more complex relationship between terms.
• 🔢 Determine the Type of Sequence: Is it arithmetic, geometric, quadratic, or something else? This dictates the general form of the formula.
• 📝 Find the Initial Term(s): The first term (usually denoted as $a_1$) is often a key component of the explicit formula.
• ✏️ Express the Pattern Algebraically: Represent the relationship between the term number ($n$) and the term value ($a_n$) using algebraic symbols and operations.
• 💡 Test Your Formula: Substitute different values of $n$ into the formula and check if the result matches the corresponding term in the sequence.

➕ Arithmetic Sequences

In an arithmetic sequence, each term is obtained by adding a constant difference ($d$) to the previous term. The explicit formula is:

$a_n = a_1 + (n - 1)d$

Example: Consider the arithmetic sequence 2, 5, 8, 11, ... Here, $a_1 = 2$ and $d = 3$. The explicit formula is:

$a_n = 2 + (n - 1)3 = 3n - 1$

✖️ Geometric Sequences

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio ($r$). The explicit formula is:

$a_n = a_1 * r^{(n - 1)}$

Example: Consider the geometric sequence 3, 6, 12, 24, ... Here, $a_1 = 3$ and $r = 2$. The explicit formula is:

$a_n = 3 * 2^{(n - 1)}$

✨ Quadratic Sequences

Quadratic sequences have a constant second difference. The general form is $a_n = An^2 + Bn + C$. Finding A, B, and C usually involves solving a system of equations.

Example: Consider the sequence 2, 7, 14, 23, ... The second difference is constant (2). By plugging in n = 1, 2, and 3 into the general form and solving the system of equations, we can find A, B, and C, leading to the explicit formula (the process of which is beyond the scope of this introductory guide, but will be in a later post!).

💡 Tips and Tricks

• 🧐 Look for Overlapping Patterns: Sometimes, sequences combine arithmetic and geometric properties.
• 🧪 Experiment with Different Forms: Try different algebraic expressions to see if they fit the sequence.
• 🧠 Use the First Few Terms: They provide crucial information for identifying the pattern and determining the formula.

📝 Practice Quiz

Write the explicit formula for each sequence:

1. 1, 4, 7, 10, ...
2. 2, 6, 18, 54, ...
3. 5, 11, 17, 23, ...
4. 10, 20, 40, 80, ...
5. -1, 1, 3, 5, ...

Answers:

1. $a_n = 3n - 2$
2. $a_n = 2 * 3^{(n - 1)}$
3. $a_n = 6n - 1$
4. $a_n = 10 * 2^{(n - 1)}$
5. $a_n = 2n - 3$

📈 Conclusion

Writing explicit formulas from sequence patterns requires careful observation, algebraic manipulation, and a bit of practice. By mastering these skills, you can unlock the secrets hidden within sequences and gain a deeper understanding of mathematical relationships.