Sequences and Series quiz

📚 Quick Study Guide

• 🔢 Arithmetic Sequence: A sequence where the difference between consecutive terms is constant. Formula: $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.
• 📈 Geometric Sequence: A sequence where each term is multiplied by a constant to get the next term. Formula: $a_n = a_1 * r^(n-1)$, where $r$ is the common ratio.
• ➕ Arithmetic Series: The sum of the terms in an arithmetic sequence. Formula: $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.
• ➗ Geometric Series: The sum of the terms in a geometric sequence. Formula: $S_n = \frac{a_1(1 - r^n)}{1 - r}$ (if $r ≠ 1$).
• ♾️ Infinite Geometric Series: The sum of an infinite geometric sequence, converging only when $|r| < 1$. Formula: $S = \frac{a_1}{1 - r}$.
• ❗ Factorial: Denoted by $n!$, is the product of all positive integers less than or equal to n. For example, $5! = 5 × 4 × 3 × 2 × 1 = 120$.
• 📊 Sigma Notation: A concise way to represent the sum of a series using the Greek letter sigma ($\Sigma$). For example, $\sum_{i=1}^{n} i$ represents the sum of integers from 1 to n.

📝 Practice Quiz

1. What is the 10th term of the arithmetic sequence 2, 5, 8, 11, ...?
   1. 29
   2. 32
   3. 35
   4. 38
2. What is the sum of the first 6 terms of the geometric sequence 3, 6, 12, 24, ...?
   1. 189
   2. 192
   3. 195
   4. 201
3. Find the sum of the first 20 terms of the series: $\sum_{n=1}^{20} (2n + 1)$
   1. 420
   2. 430
   3. 440
   4. 460
4. What is the common ratio of the geometric sequence 4, -8, 16, -32, ...?
   1. -2
   2. 2
   3. -4
   4. 4
5. Determine the sum of the infinite geometric series: 1 + 1/3 + 1/9 + 1/27 + ...
   1. 3/2
   2. 2/3
   3. 4/3
   4. 3/4
6. Evaluate 6! / 4!
   1. 30
   2. 15
   3. 360
   4. 2
7. The 3rd term of an arithmetic progression is 16 and the 7th term is 40. Find the common difference.
   1. 4
   2. 6
   3. 8
   4. 10