Algebra 2 Practice Quiz: Explicit Formulas for Arithmetic Sequences

📚 Topic Summary

An explicit formula allows you to directly calculate any term in an arithmetic sequence using its position. For an arithmetic sequence with first term $a_1$ and common difference $d$, the $n^{th}$ term, denoted as $a_n$, is given by the formula: $a_n = a_1 + (n-1)d$. This formula eliminates the need to find all the preceding terms to find a specific term, making it a powerful tool.

Let's test your knowledge with the following activity.

🧠 Part A: Vocabulary

Match each term with its definition:

1. Arithmetic Sequence
2. Common Difference
3. Explicit Formula
4. Term
5. $n^{th}$ term

Definitions (Unordered):

1. A sequence where the difference between consecutive terms is constant.
2. A formula that allows direct calculation of any term in a sequence.
3. Each number in a sequence.
4. The constant difference between consecutive terms in an arithmetic sequence.
5. The term at position 'n' in a sequence.

✏️ Part B: Fill in the Blanks

An ________ formula lets you find the value of a term in an arithmetic sequence using its ________. In the formula $a_n = a_1 + (n-1)d$, $a_1$ represents the ________ term, $n$ represents the term ________, and $d$ represents the ________ ________.

💡 Part C: Critical Thinking

Explain in your own words why using an explicit formula is more efficient than listing out terms to find $a_{100}$ in an arithmetic sequence.