Sequences and Recursion

A sequence is an ordered list of numbers, like $3, 5, 7, 9, \dots$. Sequences describe anything that happens in steps: savings each month, seats in each row, the height of each bounce. You can describe a sequence with a formula for any term, or with a rule that builds each term from the one before.

Key ideas

Terms and notation

The numbers in a sequence are its terms. We write $t_1$ for the first term, $t_2$ for the second, and $t_n$ for the $n$th term (the general term). The subscript $n$ is the term’s position: $1, 2, 3, \dots$

Sequences are discrete functions

A sequence is a function whose inputs are the term numbers $n = 1, 2, 3, \dots$ So you can also write $t_n = f(n)$. For example, $t_n = 2n + 1$ is the same as $f(n) = 2n + 1$ with $n$ a natural number.

Its graph is a set of separate points, not a joined line. A function like this, defined only at separate values, is a discrete function. A function defined for every real number in an interval, like $f(x) = 2x + 1$, is continuous.

The sequence is only the dots. The dashed line, $y = 2x + 1$, is the continuous function they lie on.

Two ways to describe a sequence

• General term (explicit formula): gives any term directly from $n$. Example: $t_n = 2n + 1$, so $t_{50} = 101$.
• Recursion formula: gives the first term, plus a rule for getting each term from the previous one. Example: $t_1 = 3$, $t_n = t_{n - 1} + 2$.

A recursion formula always needs a starting value. “Add $2$ each time” doesn’t tell you where to start.

Worked examples

Example 1: From a general term

For $t_n = 3n - 2$, find the first four terms and $t_{20}$.

Solution. Substitute $n = 1, 2, 3, 4$:

$$t_1 = 1, \quad t_2 = 4, \quad t_3 = 7, \quad t_4 = 10$$ $$t_{20} = 3(20) - 2 = 58$$

Example 2: Which term is it?

For $t_n = n^2 + 1$, find the first four terms. Which term equals $101$?

Solution. The first four terms are $2, 5, 10, 17$.

$$n^2 + 1 = 101 \quad\Rightarrow\quad n^2 = 100 \quad\Rightarrow\quad n = 10$$

($n = -10$ is also a solution of the equation, but term numbers must be positive.) So $t_{10} = 101$.

Example 3: From a recursion formula

Find the first five terms of the sequence with $t_1 = 5$ and $t_n = 2t_{n - 1} - 3$.

Solution. Each term is double the previous term, minus $3$:

$$\begin{aligned} t_1 &= 5 \\ t_2 &= 2(5) - 3 = 7 \\ t_3 &= 2(7) - 3 = 11 \\ t_4 &= 2(11) - 3 = 19 \\ t_5 &= 2(19) - 3 = 35 \end{aligned}$$

Example 4: Writing both formulas

For the sequence $4, 9, 14, 19, \dots$, write a recursion formula and a general term.

Solution. Each term is $5$ more than the one before, so a recursion formula is:

$$t_1 = 4, \qquad t_n = t_{n - 1} + 5$$

For the general term, the terms go up by $5$, so try $5n$: that gives $5, 10, 15, 20$, which is $1$ too big each time. So $t_n = 5n - 1$, or in function notation, $f(n) = 5n - 1$.

Check: $t_4 = 5(4) - 1 = 19$. ✓

Common mistakes

Starting at $n = 0$. Term numbers start at $1$, so the first term is $t_1$.

Mixing up $n$ and $t_n$. $n$ is the position; $t_n$ is the value. In Example 2, the answer is “the $10$th term”, not "$101$".

Giving a recursion formula without a first term. $t_n = t_{n - 1} + 5$ fits $4, 9, 14, \dots$ and also $1, 6, 11, \dots$. Always state $t_1$.

Joining the dots. A sequence’s graph is separate points. There’s no “term number $2.5$”.

Dropping brackets with negative signs. For $t_n = (-1)^n n^2$, the signs alternate: $t_1 = -1$ and $t_2 = 4$.

Practice

1. (Warm-up) Find the first four terms of $t_n = 4n + 3$.

Solution

$7, 11, 15, 19$

2. (Warm-up) For $t_n = 2^n - 1$, find $t_5$.

Solution

$t_5 = 2^5 - 1 = 31$

3. (Warm-up) Find the first five terms of the sequence with $t_1 = 2$ and $t_n = t_{n - 1} + 6$.

Solution

$2, 8, 14, 20, 26$

4. (Core) Find the first four terms of the sequence with $t_1 = 1$ and $t_n = 3t_{n - 1} + 1$.

Solution

$t_2 = 3(1) + 1 = 4$, $t_3 = 3(4) + 1 = 13$, $t_4 = 3(13) + 1 = 40$. So: $1, 4, 13, 40$.

5. (Core) For $6, 11, 16, 21, \dots$, write a recursion formula and a general term.

Solution

Recursion: $t_1 = 6$, $t_n = t_{n - 1} + 5$.

General term: $t_n = 5n + 1$. (Check: $t_1 = 6$, $t_4 = 21$. ✓)

6. (Core) Which term of $t_n = 7n - 4$ equals $157$?

Solution

$7n - 4 = 157$, so $7n = 161$ and $n = 23$. It’s the $23$rd term.

7. (Core) The Fibonacci sequence is defined by $t_1 = 1$, $t_2 = 1$, and $t_n = t_{n - 1} + t_{n - 2}$. Write its first ten terms.

Solution

Each term is the sum of the two before it:

$$1, 1, 2, 3, 5, 8, 13, 21, 34, 55$$

8. (Core) Find the first four terms of $t_n = (-1)^n n^2$.

Solution

$t_1 = -1$, $t_2 = 4$, $t_3 = -9$, $t_4 = 16$.

9. (Challenge) Find a general term for $2, 6, 12, 20, 30, \dots$

Solution

Write each term as a product of consecutive whole numbers: $1 \times 2,\ 2 \times 3,\ 3 \times 4,\ 4 \times 5,\ 5 \times 6$. So $t_n = n(n + 1)$.

10. (Challenge) A sequence has $t_n = 2t_{n - 1} + 1$ and $t_4 = 31$. Find $t_1$.

Solution

Work backwards: if $t_n = 2t_{n - 1} + 1$, then $t_{n - 1} = \dfrac{t_n - 1}{2}$.

$$t_3 = \frac{31 - 1}{2} = 15, \qquad t_2 = \frac{15 - 1}{2} = 7, \qquad t_1 = \frac{7 - 1}{2} = 3$$