# Introduction to Sequences

## (Notation and Terminology)

Sequences of numbers are lists of numbers. For instance, both: $3,7,11,15,19,23, \dots$

and
$1,1,2,3,5,8,13, \dots$ are sequences of numbers.

We'll often be required to predict the next few terms of a sequence, or we'll need to find a formula for its $$n^{\text{th}}$$ term.

## Notation & Terminology

To refer to any term of a sequence, we use the $$u_n$$ notation, where $$n$$ indicates the term we're referring to.

For instance, if we're dealing with the sequence $$3,7,11,15,19,23, \dots$$ we would refer to the first, second and third terms as: $u_1 = 3$ $u_2 = 7$ $u_3 = 11$ We'll often refer to the $$n^{\text{th}}$$ term of a sequence, $$u_n$$.
The $$n^{\text{th}}$$ term is the generic term of the sequence and is usually equal to some formula with $$n$$, which allows us to calculate any term of the sequence.

## Exercise 1

1. Given the sequence whose first few terms are: $2,5,8,11,14, \dots$
1. State the values of $$u_2$$ and $$u_4$$.
2. Write the values of $$u_6$$, $$u_7$$ and $$u_8$$.
2. Given the sequence whose first few terms are: $3,6,12,24,48, \dots$
1. State the values of $$u_3$$ and $$u_5$$.
2. Write the values of $$u_6$$, $$u_7$$ and $$u_8$$.

1. $$u_2 = 5$$ and $$u_4 = 11$$
2. $$u_6 = 12$$, $$u_7 = 20$$ and $$u_8 = 23$$.

1. $$u_3 = 12$$ and $$u_5 = 48$$
2. $$u_6 = 96$$, $$u_7 = 192$$ and $$u_8 = 384$$.

## Formula for the $$n^{\text{th}}$$ term

Many sequences have a formula, which allows us to calculate any term of the sequence directly.
For instance, the sequence whose first few terms are: $3,7,11,15,19,23, \dots$ has a formula: $u_n = 4n-1$ With this we can calculate any term of the sequence directly.
For example, we could check that the third term is indeed $$11$$ by replacing every $$n$$ we see in $$u_n = 4n-1$$ by $$3$$ and calculating: \begin{aligned} u_3 &= 4\times 3-1 \\ & = 12 - 1\\ u_3 & = 11 \end{aligned} Or we could even calculate the $$40^{\text{th}}$$ term. Again all we would do is replace every $$n$$ by $$40$$ and calculating: \begin{aligned} u_{40} &= 4\times 40-1 \\ & = 160 - 1\\ u_{40} & = 159 \end{aligned}

### Tutorial: Formula for the n-th term

In the following tutorial we learn about the formula for the $$n^{\text{th}}$$ term of a sequence and learn how they can be used.

## Exercise 2

1. A sequence has formula $$u_n = 3n+2$$.
Calculate:
1. Its first term, $$u_1$$.
2. Its eigth term, $$u_8$$.
3. Its fiftieth term, $$u_{50}$$.

2. A sequence has formula $$u_n = n^2+2n -3$$.
Calculate:
1. Its second term, $$u_2$$.
2. Its fifth term, $$u_5$$.
3. Its tenth term, $$u_{10}$$.

3. A sequence has formula, $$u_n = 3\times 2^{n-1}$$.
Calculate:
1. The first five terms of this sequence.
2. The eleventh term, $$u_{11}$$, of the sequence.

1. For the sequence $$u_n = 3n+2$$:
1. $$u_1 = 5$$
2. $$u_8 = 26$$
3. $$u_{50} = 152$$

2. For the sequence $$u_n = n^2 + 2n - 3$$:
1. $$u_2 = 5$$
2. $$u_5 = 32$$
3. $$u_{10} = 117$$

3. For the sequence $$u_n = 3\times 2^{n-1}$$:
1. The first $$5$$ terms of the sequence are: $u_1 = 3, \ u_2 = 6, \ u_3 = 12, \ u_4 = 24, \ u_5 = 48$
2. $$u_{11} = 3072$$