📚 Quick Study Guide
- 🔢 Sequences Defined: A sequence is an ordered list of numbers (terms), often following a specific pattern or rule. Each number in the sequence is called a term.
- ➕ Arithmetic Sequences:
- 💡 Definition: A sequence where the difference between consecutive terms is constant. This constant difference is called the common difference ($d$).
- ✍️ General Term: The formula to find any term ($T_n$) in an arithmetic sequence is $T_n = a + (n-1)d$, where $a$ is the first term, $n$ is the term number, and $d$ is the common difference.
- 📈 Sum of $n$ Terms ($S_n$): The sum of the first $n$ terms is given by $S_n = \frac{n}{2}(a + T_n)$ or $S_n = \frac{n}{2}(2a + (n-1)d)$.
- ✖️ Geometric Sequences:
- 🌟 Definition: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$).
- 📝 General Term: The formula to find any term ($T_n$) in a geometric sequence is $T_n = ar^{n-1}$, where $a$ is the first term, $n$ is the term number, and $r$ is the common ratio.
- 📊 Sum of $n$ Terms ($S_n$): The sum of the first $n$ terms is given by $S_n = \frac{a(r^n - 1)}{r - 1}$ (for $r \ne 1$) or $S_n = \frac{a(1 - r^n)}{1 - r}$ (for $r \ne 1$).
- ♾️ Sum to Infinity ($S_\infty$): For a geometric sequence to have a sum to infinity, the common ratio $r$ must satisfy $|r| < 1$. The formula is $S_\infty = \frac{a}{1 - r}$.
- 🧠 Key Steps for Problem Solving:
- 🧐 Identify Type: Determine if the sequence is arithmetic, geometric, or neither.
- 🔍 Find $a, d,$ or $r$: Identify the first term, common difference, or common ratio.
- 🎯 Apply Formula: Use the appropriate formula for the general term or sum.
✅ Practice Quiz
Question 1: What is the 8th term of the arithmetic sequence 3, 7, 11, ...?
- 31
- 35
- 28
Question 2: Find the sum of the first 10 terms of the arithmetic sequence 2, 5, 8, ...
- 155
- 145
- 160
Question 3: What is the common ratio of the geometric sequence 5, 15, 45, ...?
- 3
- 10
- \frac{1}{3}
Question 4: Find the 5th term of the geometric sequence where $a=4$ and $r=2$.
- 32
- 64
- 16
Question 5: Calculate the sum of the first 6 terms of the geometric sequence 2, 6, 18, ...
- 728
- 242
- 364
Question 6: For which of the following sequences does a sum to infinity exist?
- 3, 6, 12, ...
- 100, 50, 25, ...
- -1, -2, -4, ...
Question 7: An arithmetic sequence has $T_3 = 10$ and $T_7 = 26$. What is the first term ($a$) of the sequence?
- 2
- 4
- 6
Click to see Answers
1. A (Explanation: $a=3, d=4$. $T_8 = 3 + (8-1)4 = 3 + 7 \times 4 = 3 + 28 = 31$)
2. A (Explanation: $a=2, d=3$. $T_{10} = 2 + (10-1)3 = 2 + 27 = 29$. $S_{10} = \frac{10}{2}(2+29) = 5(31) = 155$)
3. A (Explanation: $r = \frac{15}{5} = 3$)
4. B (Explanation: $T_5 = ar^{5-1} = 4(2)^4 = 4(16) = 64$)
5. A (Explanation: $a=2, r=3$. $S_6 = \frac{2(3^6 - 1)}{3 - 1} = \frac{2(729 - 1)}{2} = 728$)
6. B (Explanation: A sum to infinity exists if $|r|<1$. For 100, 50, 25, ..., $r=\frac{50}{100} = \frac{1}{2}$, and $|\frac{1}{2}| < 1$. Options A and C have $|r| > 1$.)
7. B (Explanation: $T_3 = a + 2d = 10$. $T_7 = a + 6d = 26$. Subtracting the first from the second: $(a+6d) - (a+2d) = 26 - 10 \implies 4d = 16 \implies d=4$. Substitute $d=4$ into $a+2d=10 \implies a+2(4)=10 \implies a+8=10 \implies a=2$.)