online sequences and series quiz

📚 Quick Study Guide

• 🔢 Arithmetic Sequences: A sequence where the difference between consecutive terms is constant. The formula for the $n$th term is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
• ➕ Geometric Sequences: A sequence where each term is multiplied by a constant to get the next term. The formula for the $n$th term is $a_n = a_1 * r^(n-1)$, where $a_1$ is the first term and $r$ is the common ratio.
• 📈 Series: The sum of the terms in a sequence.
• 🧮 Arithmetic Series: The sum of an arithmetic sequence. The formula for the sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.
• ➗ Geometric Series: The sum of a geometric sequence. The formula for the sum of the first $n$ terms is $S_n = \frac{a_1(1 - r^n)}{1 - r}$, where $r \neq 1$.
• ♾️ Infinite Geometric Series: A geometric series with an infinite number of terms. It converges (has a finite sum) only if $|r| < 1$. The sum is given by $S = \frac{a_1}{1 - r}$.
• ✍️ Sigma Notation: A concise way to represent series using the Greek letter sigma ($\sum$). For example, $\sum_{i=1}^{n} a_i$ represents the sum of the terms $a_i$ from $i = 1$ to $i = n$.

🧪 Practice Quiz

1. Which of the following sequences is an arithmetic sequence?
   1. A. 2, 4, 8, 16, ...
   2. B. 1, 4, 9, 16, ...
   3. C. 3, 7, 11, 15, ...
   4. D. 2, 6, 18, 54, ...
2. What is the common difference in the arithmetic sequence 5, 8, 11, 14, ...?
   1. A. 2
   2. B. 3
   3. C. 4
   4. D. 5
3. Find the 7th term of the geometric sequence 2, 6, 18, ...
   1. A. 486
   2. B. 1458
   3. C. 4374
   4. D. 13122
4. What is the common ratio in the geometric sequence 4, -8, 16, -32, ...?
   1. A. -2
   2. B. 2
   3. C. -4
   4. D. 4
5. Calculate the sum of the first 5 terms of the arithmetic sequence 1, 3, 5, 7, ...
   1. A. 15
   2. B. 20
   3. C. 25
   4. D. 30
6. Find the sum of the infinite geometric series 9 + 3 + 1 + 1/3 + ...
   1. A. 12.5
   2. B. 13.5
   3. C. 14.5
   4. D. 15.5
7. Evaluate the series expressed in sigma notation: $\sum_{i=1}^{4} (2i + 1)$
   1. A. 16
   2. B. 20
   3. C. 24
   4. D. 28