Arithmetic Sequences
- An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant.
- This constant difference is called the common difference, typically denoted by $d$.
Definition and Notation
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.
- Consider the sequence: 3, 7, 11, 15, 19, ...
- Here, $a_1 = 3$ and the common difference $d = 4$.
- Terms can also be represented as $a_n$ instead of $u_n$.
- The two notations are interchangeable.
Formula for the nth Term
- The general formula for the $n$th term of an arithmetic sequence is: $$u_n = u_1 + (n-1)d$$ where
- $u_1$ is the first term, $n$ is the position of the term
- $d$ is the common difference
- This is because for every term, you're adding $d$ to the previous term.
- To get to the $n$th term from the first term, you perform $n - 1$ additions of $d$.
To find the common difference quickly, subtract any term from the subsequent term: $d = a_{n+1} - a_n$
For the sequence 3, 7, 11, 15, 19, ..., find the 10th term.
Solution
$$a_1 = 3 \qquad d = 4 $$
$$a_{10} = 3 + (10-1)4 = 3 + 36 = 39$$
Arithmetic Series
- An arithmetic series is the sum of the terms of an arithmetic sequence.
- This is denoted by $S_n$, where $n$ is the number of terms you are summing together.
Formulas for arithmetic series
- Let's take the first and last terms of an arithmetic sequence, or $u_1$ and $u_n$.
- The sum of these is $(u_1 + u_n)$.
- Now, let's take the second and second-to-last terms of an arithmetic sequence, or $u_2$ and $u_{n-1}$.
- These can be expressed as $u_1 + d$ and $u_n - d$ respectively.
- Therefore, $u_2 + u_{n-1} = u_1 + d + u_n - d = u_1 + u_n$.
- You can repeat this for all $\frac{n}2$ pairs of $u_k$ and $u_{n - k}$ going down to the middle.
- If $n$ is odd, this still works because $u_{\frac{n}2} = \frac{u_n}2$.
- Thus, the sum of the first $n$ terms of an arithmetic sequence is given by: $$S_n = \frac{n}{2}(u_1 + u_n)$$
- 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, ...
Solution
$$u_1 = 3 \qquad d = 4 \qquad n = 20 \qquad u_{20} = 3 + (20-1)4 = 79$$
Using the first formula: $$S_{20} = \frac{20}{2}(3 + 79) = 10(82) = 820$$
Sigma Notation
- Sigma notation is a compact way to represent sums.
- For arithmetic sequences, it's particularly useful.
- The sum of an arithmetic sequence can be written as: $$\sum_{k=1}^n (u_1 + (k-1)d) = u_1 + (u_1 + d) + (u_1 + 2d) + ... + (u_1 + (n-1)d)$$
- The index $k$ typically starts at 1, but it can start at any value.
- The upper limit $n$ determines how many terms are summed.
Express the sum of the first 50 positive integers using sigma notation.
Solution
$$\sum_{i=1}^{50} i$$
This is equivalent to $1 + 2 + 3 + ... + 49 + 50$
Applications
Simple Interest
- Simple interest is a common application of arithmetic sequences.
- Simple interest is interest where a bank balance increases by a set percentage of the initial amount every year.
- If an initial amount $P$ is invested at an annual simple interest rate $r$ for $n$ years, the interest $I$ is given by: $$I = P \cdot r \cdot n$$
- 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?
Solution
$$P = 1000 \qquad r = 0.05 \qquad n = 10$$
$$A = 1000(1 + 0.05 \cdot 10) = 1000(1.5) = \text{\$}1500$$
