Sequences and Series
Class 11 Mathematics · Chapter 8
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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: