Chapter 8 ยท Introduction

Chapter 8 โ€” Sequences and Series

Lists of numbers that follow a rule โ€” savings that grow, medicines that decay, rabbits that multiply. Name the $n$th term, add terms with $\Sigma$, master the geometric progression and compare the two famous averages.

An old Indian story tells of the minister who invented the game of chess and was offered any reward he wished by a delighted king. He asked only for rice: one grain on the first square of the board, two on the second, four on the third, and so on, doubling each time up to the sixty-fourth square. The king laughed at so small a request โ€” until his treasurers worked it out. The last square alone needs $2^{63}$ grains, and the whole board $2^{64}-1=18\,446\,744\,073\,709\,551\,615$ grains: hundreds of times the rice the whole world now harvests in a year. Doubling, repeated, is far more powerful than it looks.

Another Indian thread runs through this chapter. Poets studying rhythms of short and long syllables โ€” Virahฤแน…ka more than a thousand years ago and Hemachandra in the twelfth โ€” found that the number of rhythm patterns of each length is the sum of the two before it: $1,2,3,5,8,13,\dots$, the numbers Europe later named after Fibonacci. In this chapter you will learn the language for such lists: a sequence is a list that follows a rule, a series is what you get when you add its terms, and the sign $\Sigma$ writes long sums in a few symbols. You met the arithmetic progression in Class 10, where each term is the one before plus a fixed number. Here the star is the geometric progression, where each term is the one before times a fixed number โ€” the pattern behind compound interest, population growth, radioactive decay and the king's rice. Finally you will compare the two natural averages of two numbers, the arithmetic mean and the geometric mean, and prove that the first is never smaller than the second.

Sequence: a rule for the $n$th termโ†’Series and $\Sigma$ notationโ†’GP: $a_n=ar^{n-1}$, $S_n$, $S_\infty$โ†’Means: A.M. $\ge$ G.M.
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • describe a sequence by a formula for its $n$th term or by a recursive rule (such as the Fibonacci rule), and write down or find any term;
  • write a series in $\Sigma$ notation, expand a $\Sigma$ back into a sum, and use the rules for splitting sums and $a_n=S_n-S_{n-1}$;
  • recognise a geometric progression, find its $n$th term, the sum of its first $n$ terms and โ€” when $|r|\lt1$ โ€” the sum of all its infinitely many terms;
  • solve problems on compound interest, depreciation, population growth and recurring decimals with geometric progressions;
  • find the arithmetic mean and geometric mean of two numbers, insert any number of arithmetic or geometric means between two numbers, and prove and use the inequality A.M. $\ge$ G.M.
Teaser โ€” Adding a Fixed Amount versus Multiplying by a Fixed Ratio

Two people each invest โ‚น10 000 at the same rate. Plan A pays simple interest โ€” the same amount is added every year, so the yearly values form an arithmetic progression (blue). Plan B pays compound interest โ€” the value is multiplied by the same factor every year, a geometric progression (red). Press Play to let the years run, change the rate, and watch who pulls ahead and how fast.

10%30
year $n$โ€”
simple (AP)โ€”
compound (GP)โ€”
doubling timeโ€”

Sections in this chapter

8.1

Sequences and Series

Sequences as rules, finite and infinite sequences, recursive definitions and the Fibonacci numbers, series and $\Sigma$ notation, and a quick recall of the arithmetic progression.

$S_n=\displaystyle\sum_{k=1}^{n}a_k$
Open section โ†’8.2

Geometric Progressions

The common ratio, the $n$th term, the sum of $n$ terms, the sum to infinity when $|r|\lt1$, and applications to interest, growth, decay and recurring decimals.

$a_n=ar^{n-1},\ \ S_\infty=\dfrac{a}{1-r}$
Open section โ†’8.3

Arithmetic and Geometric Means

The A.M. and G.M. of two numbers, inserting means between two numbers, and the inequality A.M. $\ge$ G.M. with its uses in maximum and minimum problems.

$\dfrac{a+b}{2}\ge\sqrt{ab}$
Open section โ†’
โœ… Before you start
From Class 10 Mathematics you need arithmetic progressions (Chapter 5): the common difference $d$, the $n$th term $a_n=a+(n-1)d$ and the sum $S_n=\tfrac n2\,[2a+(n-1)d]$ โ€” Section 8.1 recalls them in one paragraph and then builds on them. You need the laws of exponents ($a^m a^n=a^{m+n}$, $(a^m)^n=a^{mn}$, $a^{-n}=1/a^n$) and comfort with percentages for interest and growth problems, and solving quadratic equations (Class 10, Chapter 4) for finding unknown terms. The idea of a function from Chapter 2 of this course appears at the very start: a sequence is a function whose inputs are $1,2,3,\dots$ . The binomial theorem of Chapter 7 is not needed, though you will meet the numbers $2^n$ and powers like $(1.1)^n$ from both directions.
โ† 7.2 Applications of the Binomial Expansion8.1 Sequences and Series โ†’