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Sequences and Series
Introduction
Sequences
Series
Practice: Terms of a Sequence
Geometric Progression (G. P.)
Relationship Between A.M. and G.M.
Practice: Geometric Progressions
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Glossary

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Sequences and SeriesSequences

Reading time: ~20 min

Here is a question about your own family. You have 2 parents, 4 grandparents, 8 great-grandparents, and so on further back. Suppose a new generation is born every 30 years. How many generations back can you go in 300 years, and how many ancestors are there in each of those generations?

300 years at 30 years per generation gives \dfrac{300}{30} = generations.

Going back one generation at a time, the numbers of ancestors are 2, 4, 8, 16, 32, \ldots — each number in the list is the one before it.

So the tenth generation back holds 2^{10} = people.

These ten numbers, written in order, form a sequence: there is a first number, a second number, a third, and so on, up to the tenth.

Here is another sequence, from a very ordinary division. Divide 10 by 3 and write down the quotient after each step of the long division: 3,\; 3.3,\; 3.33,\; 3.333, \ldots These quotients form a sequence too — and this one never stops, because the division never ends.

Each number in a sequence is called a term. We name the terms a_1, a_2, a_3, \ldots, a_n, \ldots, where the small number written below — the subscript — tells you the term's . So a_n is the term in the nth position. It is also called the general term, or the nth term, of the sequence.

For the ancestors, a_1 = 2,\; a_2 = 4,\; a_3 = 8,\; \ldots,\; a_{10} = 1024. For the quotients, a_1 = 3,\; a_2 = 3.3,\; a_3 = 3.33, and the sixth term is a_6 = .

A sequence with a fixed number of terms is a finite sequence: the ancestors over 300 years stop at the 10th term. A sequence that is not finite — one that goes on forever, like the quotients — is an infinite sequence.

Sort these sequences: does each one stop, or does it go on forever?

the even numbers 2, 4, 6, 8, …
the dates of the month of March
the quotients 3, 3.3, 3.33, … of 10 ÷ 3
the runs scored after each over of a T20 innings
the prime numbers
the multiples of 7 that are less than 1000
the powers of 2: 2, 4, 8, 16, …
your ancestors over the last 300 years, generation by generation
Finite
Infinite

A quick test: if you could, at least in principle, reach a last term, the sequence is finite. If there is always a next term, it is infinite.

Many sequences come with a formula that produces any term you ask for. Take the even natural numbers 2, 4, 6, 8, \ldots:

\begin{aligned} a_1 &= 2 \times 1 = 2 \\ a_2 &= 2 \times 2 = 4 \\ a_3 &= 2 \times 3 = 6 \\ &\;\;\vdots \\ a_{23} &= 2 \times 23 = 46 \end{aligned}

So the nth term is a_n = , for every natural number n. In the same way, the odd natural numbers 1, 3, 5, \ldots have a_n = .

Not every sequence has a formula like that. The sequence 1, 1, 2, 3, 5, 8, \ldots is built by a recurrence relation: each new term is made from the terms before it.

a_1 = a_2 = 1, \qquad a_n = a_{n-2} + a_{n-1} \quad (n > 2)

This is the Fibonacci sequence: every term is the sum of the two before it. And the primes 2, 3, 5, 7, 11, \ldots have no formula at all — the only way to describe them is in words, "the primes, in increasing order". So we do not insist on a formula. What every sequence does need is a clear rule that decides which number sits in each position.

That is why a sequence can be seen as a function whose domain is the set of natural numbers, or a part of it such as \{1, 2, \ldots, k\}: feed in a position n, and out comes the term a_n. Sometimes we even write a(n) instead of a_n.

The term machine: pick a rule, set the position n, and read off the term that comes out. Complete all three missions.

Position n = ${n}

A formula, a recurrence or a description in words — each one is a rule that turns a position into a term, which is exactly what a function does.

Speed Round: Spot the Rule

Each card shows the first four terms of a sequence. Pick the formula for its general term a_n. Test n = 1 and n = 2 in your head before you tap — and be careful, some wrong formulas get the first term right.

Get five right to move on.

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