Sequences and series

Num­bers often come in lists that fol­low a rule: the odd num­bers, the seats in the rows of a the­atre, the money in a sav­ings account month by month. Such a list is called a sequence, and adding up its terms gives a series. This les­son sets out the words and nota­tion used for the whole chap­ter, so that arith­metic, geo­met­ric and har­monic pro­gres­sions can be described pre­cisely in the lessons that fol­low.

Sequence#

A set of num­bers arranged in a def­i­nite order accord­ing to some def­i­nite rule is called a sequence.

For exam­ple, $2, 4, 6, 8, 10, \ldots$ is a sequence: the rule is "the even num­bers in increas­ing order". So is $1, 4, 9, 16, 25, \ldots$, the squares of the nat­ural num­bers.

The num­bers in a sequence are called its terms. We write the first term as $t_1$, the sec­ond as $t_2$, and so on. The term in posi­tion $n$ is $t_n$, called the $n$th term or gen­eral term. (Some books write $a_n$ instead of $t_n$; the mean­ing is the same.)

The impor­tant thing is that the order mat­ters. The num­bers $1, 3, 5$ and $5, 3, 1$ are dif­fer­ent sequences, even though they con­tain the same num­bers.

Describ­ing a sequence by its gen­eral term#

The neat­est way to give a sequence is a for­mula for $t_n$. Put $n = 1, 2, 3, \ldots$ to get the terms one after another.

Exam­ple 1#

Write the first four terms of the sequence with $t_n = 3n - 1$.

$$t_1 = 3(1) - 1 = 2, \quad t_2 = 3(2) - 1 = 5, \quad t_3 = 3(3) - 1 = 8, \quad t_4 = 3(4) - 1 = 11$$

So the sequence is $2, 5, 8, 11, \ldots$

Exam­ple 2#

Write the first four terms of the sequence with $t_n = n^2 + 1$, and find $t_{10}$.

$t_1 = 2$, $t_2 = 5$, $t_3 = 10$, $t_4 = 17$. And $t_{10} = 10^2 + 1 = 101$.

Exam­ple 3#

Write the first five terms of the sequence with $t_n = (-1)^n \cdot n$.

When $n$ is odd, $(-1)^n = -1$; when $n$ is even, $(-1)^n = 1$. So the terms are $-1, 2, -3, 4, -5, \ldots$, with signs that alter­nate.

Describ­ing a sequence by a rule link­ing terms#

Some­times each term is built from the terms before it. Such a rule is called a recur­sive rule. For exam­ple:

$$t_1 = 1, \quad t_2 = 1, \quad t_n = t_{n-1} + t_{n-2} \text{ for } n \ge 3$$

Each term is the sum of the two before it, so the sequence is $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ This famous sequence is named after the Ital­ian math­e­mati­cian Fibonacci, and it turns up in the spi­rals of sun­flower seeds and pine cones.

Exam­ple 4#

If $t_1 = 3$ and $t_n = 2t_{n-1} + 1$ for $n \ge 2$, write the first four terms.

$t_1 = 3$, $t_2 = 2(3) + 1 = 7$, $t_3 = 2(7) + 1 = 15$, $t_4 = 2(15) + 1 = 31$.

Finite and infi­nite sequences#

A sequence that stops after a cer­tain num­ber of terms is finite, for exam­ple the two-digit even num­bers $10, 12, 14, \ldots, 98$. A sequence that goes on for ever is infi­nite, for exam­ple $\displaystyle 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$ The three dots at the end of a list mean that the terms con­tinue in the same way.

Is a given num­ber a term?#

To decide whether a num­ber belongs to a sequence, set $t_n$ equal to it and solve for $n$. The num­ber is a term only if $n$ comes out as a nat­ural num­ber.

Exam­ple 5#

Is 100 a term of the sequence $t_n = 3n + 1$? Is 50?

$3n + 1 = 100$ gives $3n = 99$ and $n = 33$, a nat­ural num­ber. So 100 is the 33rd term.

$3n + 1 = 50$ gives $3n = 49$ and $\displaystyle n = \frac{49}{3}$, which is not a nat­ural num­ber. So 50 is not a term.

Series#

The sum of the terms of a sequence is called a series. If $t_1, t_2, t_3, \ldots$ is a sequence, then the expres­sion

$$t_1 + t_2 + t_3 + \cdots$$

is a series. A series is called a finite series if it has a finite num­ber of terms; oth­er­wise it is called an infi­nite series.

The sum of the first $n$ terms is writ­ten $S_n$:

$$S_n = t_1 + t_2 + \cdots + t_n$$

A short way to write this uses the Greek cap­i­tal let­ter sigma: $\displaystyle S_n = \sum_{k=1}^{n} t_k$, read "the sum of $t_k$ as $k$ goes from 1 to $n$".

Exam­ple 6#

For the sequence $t_n = 2n - 1$, find $S_4$.

The first four terms are $1, 3, 5, 7$. So $S_4 = 1 + 3 + 5 + 7 = 16$.

From sums back to terms#

If you know the sum of the first $n$ terms for every $n$, you can recover each term. The first $n$ terms add up to $S_n$, and the first $n - 1$ terms add up to $S_{n-1}$; the dif­fer­ence is the $n$th term alone:

$$t_n = S_n - S_{n-1} \quad (n \ge 2), \qquad t_1 = S_1$$

Exam­ple 7#

The sum of the first $n$ terms of a sequence is $S_n = n^2 + 2n$. Find the first three terms.

$t_1 = S_1 = 1 + 2 = 3$. Then $S_2 = 4 + 4 = 8$, so $t_2 = S_2 - S_1 = 8 - 3 = 5$. And $S_3 = 9 + 6 = 15$, so $t_3 = 15 - 8 = 7$. The sequence begins $3, 5, 7, \ldots$

Prac­tice#

1. Write the first four terms of the sequence with $t_n = 5n + 2$.
2. Write the first four terms of the sequence with $\displaystyle t_n = \frac{n}{n + 1}$.
3. Find $t_8$ for the sequence with $t_n = n^2 - 3n$.
4. Write the first five terms of $t_n = (-1)^{n+1} \cdot 2n$.
5. If $t_1 = 2$ and $t_n = 3t_{n-1} - 1$ for $n \ge 2$, write the first four terms.
6. Is 77 a term of the sequence $t_n = 4n - 3$? Is 90?
7. Clas­sify as finite or infi­nite: (a) the mul­ti­ples of 5 between 10 and 100; (b) $1, 2, 4, 8, \ldots$
8. For the sequence $t_n = 3n$, find $S_5$.
9. If $S_n = 2n^2 + n$, find $t_1$, $t_2$ and $t_3$.
10. If $S_n = n^2 + 4n$, find a for­mula for $t_n$.

Answers#