What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

Define a sequence and its general term.

A sequence is an ordered arrangement of numbers following a rule; the \(n\)th or general term \(a_n\) gives the number in the \(n\)th position.

What is a geometric progression?

A sequence in which the ratio of any term to its preceding term is a constant \(r\) (the common ratio), with each term non-zero.

Write the \(n\)th term and the sum of \(n\) terms of a G.P.

\(a_n=ar^{n-1}\); and \(S_n=\dfrac{a(r^n-1)}{r-1}\) when \(r\ne1\).

Find the 10th term of the G.P. \(5,25,125,\dots\)

Here \(a=5,\ r=5\), so \(a_{10}=5\cdot5^{9}=5^{10}\).

Define the geometric mean of two positive numbers.

The geometric mean of \(a\) and \(b\) is \(\sqrt{ab}\); the three numbers \(a,\sqrt{ab},b\) form a G.P.

State the relationship between A.M. and G.M.

For two positive numbers, \(\text{A.M.}=\dfrac{a+b}{2}\ge\sqrt{ab}=\text{G.M.}\), with equality when \(a=b\).

Which term of the G.P. \(2,8,32,\dots\) is \(131072\)?

With \(a=2,\ r=4\): \(131072=2\cdot4^{n-1}\Rightarrow 4^{n-1}=65536=4^8\Rightarrow n=9\).

A G.P. has 3rd term 24 and 6th term 192. Find the 10th term.

From \(ar^2=24\) and \(ar^5=192\), \(r^3=8\Rightarrow r=2,\ a=6\); so \(a_{10}=6\cdot2^9=3072\).

Insert three numbers between 1 and 256 to form a G.P.

With \(r^4=256\Rightarrow r=4\): the numbers are \(4,16,64\).

Find A.M. and G.M. of two numbers if they are 10 and 8.

A.M. \(=\dfrac{a+b}{2}=10\) and G.M. \(=\sqrt{ab}=8\) give \(a+b=20,\ ab=64\), so the numbers are 4 and 16.

Sequences and Series

Class 11 Mathematics · Chapter 8

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Every formula of Ch 1–14, with its condition

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25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

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Summary

A sequence is an ordered list of numbers \(a_1,a_2,a_3,\dots\) whose terms follow a rule, often a formula for the \(n\)th (general) term \(a_n\) or a recurrence such as the Fibonacci relation \(a_n=a_{n-1}+a_{n-2}\). A sequence is finite or infinite; the associated series is the sum \(a_1+a_2+\cdots\), written compactly with sigma notation \(\sum_{k=1}^{n}a_k\). The chapter focuses on the geometric progression (G.P.), in which each term bears a constant ratio \(r\) to its predecessor, so \(a_n=ar^{n-1}\). The sum of the first \(n\) terms is \(S_n=\dfrac{a(r^n-1)}{r-1}\) for \(r\ne1\) (and \(S_n=na\) when \(r=1\)). Between two positive numbers \(a,b\) the geometric mean is \(\sqrt{ab}\), and we can insert several geometric means to form a G.P. For two positive numbers the arithmetic and geometric means satisfy \(\text{A.M.}\ge\text{G.M.}\). Between two positive numbers we can also insert any number of geometric means \(G_1,G_2,\dots,G_n\) so that \(a,G_1,\dots,G_n,b\) is a G.P. with \(r=(b/a)^{1/(n+1)}\). These results model growth such as compound interest, doubling populations of bacteria and the count of ancestors over successive generations, and they connect with the arithmetic mean studied earlier through the inequality \(\text{A.M.}\ge\text{G.M.}\).

Sequences and the general termSeries and sigma notationGeometric progression and its \(n\)th termSum of \(n\) terms of a G.P.Geometric meanRelationship between A.M. and G.M.

Key terms

Sequence

An ordered list of numbers \(a_1,a_2,\dots\) following a rule; its \(n\)th term is \(a_n\).

Series

The sum of the terms of a sequence, written \(\sum_{k=1}^{n}a_k\).

Geometric progression

A sequence in which each term is a constant ratio \(r\) times the previous one.

General term of a G.P.

\(a_n=ar^{n-1}\), with first term \(a\) and common ratio \(r\).

Sum of a G.P.

\(S_n=\dfrac{a(r^n-1)}{r-1}\) for \(r\ne1\); \(S_n=na\) for \(r=1\).

Geometric mean

The G.M. of \(a,b>0\) is \(\sqrt{ab}\); and \(\text{A.M.}\ge\text{G.M.}\).

Extra questions & answers

Try one from this chapter

The \(n\)th term of a sequence is denoted by:

Flashcards & interactive practice