等差与等比数列

Introduction: Two Ways to Define a Sequence

Position Formula and Recurrence

What are the first 3 terms of the sequence given by $a_k = k^2$?

Definition of Arithmetic Progression

What is the common difference of $10,8,6,4,\ldots$?

General Term and Sum of an AP

For an AP with $a_1=2$ and $r=3$, what is $a_5$?

Definition of Geometric Progression

What is the common ratio of $3,6,12,24,\ldots$?

Worked Example: Multiply by the Common Ratio and Subtract

Use the summation formula to compute $1 + 2 + 3 + \cdots + 100$. Enter an integer.

For a geometric progression with $a_1 = 2$ and $q = 3$, find $a_5$. Enter an integer.

Compute $\sum_{k=0}^{4} 2^k = 1 + 2 + 4 + 8 + 16$. Enter an integer.

What are the general-term and summation formulas for an arithmetic progression and a geometric progression?

In your own words, explain why "multiply by the common ratio and subtract" yields the sum of the first $n$ terms of a GP. Which terms cancel after the subtraction?

Comprehensive Check

Common Pitfalls and Lesson Summary