SL 1.2—Sequences and Sigma Notation Notes

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.

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, and $d$ 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 $d$.

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:

$$S_n = \frac{n}{2}(2u_1 + (n-1)d)$$

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:

$$\sum_{k=1}^n (u_1 + (k-1)d) = u_1 + (u_1 + d) + (u_1 + 2d) + ... + (u_1 + (n-1)d)$$

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:

$$A = P + I = P(1 + rn)$$