## Arithmetic Sequences [6ebbf5335798] 1. An **arithmetic sequence** is a sequence of numbers where the difference between consecutive terms is constant. 2. This constant difference is called the **common difference**, typically denoted by $d$. ### Definition and Notation [8128fa260616] For an arithmetic sequence $u_1, u_2, u_3, ...$, where $u_n$ represents the **$n$th term**: $$u_n = u_{n-1} + d$$ where $d$ is the common difference. [{"_key":"Uh9OHzbAyEX8wfy9wcBT16","_type":"block","asset":null,"children":[{"_key":"Uh9OHzbAyEX8wfy9wcBT5d","_type":"span","marks":[],"text":"Consider the sequence: 3, 7, 11, 15, 19, ..."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"Uh9OHzbAyEX8wfy9wcBTEh","_type":"block","asset":null,"children":[{"_key":"Uh9OHzbAyEX8wfy9wcBTJE","_type":"span","marks":[],"text":"Here, $a_1 = 3$ and the common difference $d = 4$."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] [{"_key":"82c78d0b61d9","_type":"block","asset":null,"children":[{"_key":"69a1d8e62351","_type":"span","marks":["strong"],"text":"Terms"},{"_key":"aca96201820b","_type":"span","marks":[],"text":" can also be represented as $a_n$ instead of $u_n$. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"add2683b02ab","_type":"block","asset":null,"children":[{"_key":"da0e74eaf760","_type":"span","marks":[],"text":"The two notations are interchangeable."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] ### Formula for the nth Term [4f47cdef3985] 1. The** general formula **for the $n$th term of an arithmetic sequence is: $$u_n = u_1 + (n-1)d$$ where 1. $u_1$ is the first term, $n$ is the position of the term 2. $d$ is the common difference 2. This is because for **every term**, you're adding $d$ to the previous term. 3. To get to the $n$th term from the first term, you perform $n - 1$ additions of $d$. [{"_key":"Uh9OHzbAyEX8wfy9wcBTNl","_type":"block","asset":null,"children":[{"_key":"Uh9OHzbAyEX8wfy9wcBTSI","_type":"span","marks":[],"text":"To find the "},{"_key":"03a040ebbbe3","_type":"span","marks":["strong"],"text":"common difference"},{"_key":"2fd9a165a533","_type":"span","marks":[],"text":" quickly, subtract any term from the subsequent term: $d = a_{n+1} - a_n$"}],"markDefs":[],"style":"normal"}] For the sequence 3, 7, 11, 15, 19, ..., find the 10th term. $$a_1 = 3 \qquad d = 4 $$ $$a_{10} = 3 + (10-1)4 = 3 + 36 = 39$$ ## Arithmetic Series [6b13a2ce8c80] 1. An **arithmetic series** is the sum of the terms of an arithmetic sequence. 2. This is denoted by $S_n$, where $n$ is the number of terms you are summing together. ### Formulas for arithmetic series [9767dee2590a] 1. Let's take the **first** and **last terms** of an arithmetic sequence, or $u_1$ and $u_n$. 2. The sum of these is $(u_1 + u_n)$. 3. Now, let's take the **second** and **second-to-last** terms of an arithmetic sequence, or $u_2$ and $u_{n-1}$. 4. These can be expressed as $u_1 + d$ and $u_n - d$ respectively. 5. Therefore, $u_2 + u_{n-1} = u_1 + d + u_n - d = u_1 + u_n$. 6. You can **repeat** this for all $\frac{n}2$ pairs of $u_k$ and $u_{n - k}$ going down to the middle. 7. If $n$ is **odd**, this still works because $u_{\frac{n}2} = \frac{u_n}2$. 8. Thus, the **sum of the first $n$ terms** of an arithmetic sequence is given by: $$S_n = \frac{n}{2}(u_1 + u_n)$$ 9. If we substitute the formula $a_n = a_1 + (n-1)d$, this becomes: $$S_n = \frac{n}{2}(2u_1 + (n-1)d)$$ Find the sum of the first 20 terms of the sequence 3, 7, 11, 15, 19, ... $$u_1 = 3 \qquad d = 4 \qquad n = 20 \qquad u_{20} = 3 + (20-1)4 = 79$$ Using the first formula: $$S_{20} = \frac{20}{2}(3 + 79) = 10(82) = 820$$ ## Sigma Notation [b159e82d562f] 1. **Sigma notation** is a compact way to represent sums. 2. For arithmetic sequences, it's particularly useful. 3. The **sum of an arithmetic sequence** can be written as: $$\sum_{k=1}^n (u_1 + (k-1)d) = u_1 + (u_1 + d) + (u_1 + 2d) + ... + (u_1 + (n-1)d)$$ [{"_key":"Uh9OHzbAyEX8wfy9wcBUTk","_type":"block","asset":null,"children":[{"_key":"Uh9OHzbAyEX8wfy9wcBUYH","_type":"span","marks":[],"text":"The "},{"_key":"c57c45e2f04e","_type":"span","marks":["strong"],"text":"index"},{"_key":"d0f585fc338b","_type":"span","marks":[],"text":" $k$ typically starts at 1, but it can start at any value. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"41296552446e","_type":"block","asset":null,"children":[{"_key":"735114a26f6e","_type":"span","marks":[],"text":"The "},{"_key":"9dd0ff0db591","_type":"span","marks":["strong"],"text":"upper limit"},{"_key":"797496f6702c","_type":"span","marks":[],"text":" $n$ determines how many terms are summed."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] Express the sum of the first 50 positive integers using sigma notation. $$\sum_{i=1}^{50} i$$ This is equivalent to $1 + 2 + 3 + ... + 49 + 50$ ## Applications [ae09d3ba5948] ### Simple Interest [a8a929d6727a] 1. **Simple interest** is a common application of arithmetic sequences. 2. Simple interest is interest where a bank balance increases by a set percentage of the **initial amount** every year. 3. If an initial amount $P$ is invested at an annual simple interest rate $r$ for $n$ years, the interest $I$ is given by: $$I = P \cdot r \cdot n$$ 4. The total amount $A$ after $n$ years is: $$A = P + I = P(1 + rn)$$ If \$1000 is invested at 5% simple interest, what will be the total amount after 10 years? $$P = 1000 \qquad r = 0.05 \qquad n = 10$$ $$A = 1000(1 + 0.05 \cdot 10) = 1000(1.5) = \text{\$}1500$$