# Sequences And Series class 11 Notes Mathematics

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## Sequences And Series class 11 Notes Mathematics

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CBSE Class 11 Mathematics
Revision Notes
Chapter-9
Sequences And Series class 11 Notes Mathematics

1. Sequences and series
2. Arithmetic Progression (A.P.)
3. Geometric Progression (G.P.), relation in A.M. and G.M.
4. Sum to n terms of Special Series

• Sequence: By a sequence, we mean an arrangement of number in definite order according to some rules. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3, ….k}. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence.
• Let ${{\text{a}}_{1}},\text{ }{{\text{a}}_{2}},\text{ }{{\text{a}}_{3}}$, … be the sequence, then the sum expressed as ${{\text{a}}_{1}}\text{+ }{{\text{a}}_{2}}\text{+ }{{\text{a}}_{3}}+.....$  is called series. A series is called finite series if it has got finite number of terms.

ARITHMETIC PROGRESSION

• An arithmetic progression (A.P .) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called  common difference of the A.P. Usually, we denote the first term of A.P . by   a, the common difference by d and the last term by $l$. The general term or the  nth  term of the A.P. is given by ${{\text{a}}_n} = {\text{ a }} + {\text{ }}\left( {{\text{n }}-{\text{ 1}}} \right){\text{ d}}.$
• Single Arithmetic mean between any two given numbers a and b: A.M. = ${{a + b} \over 2}$
• $n$ Arithmetic mean between two given numbers a and b: $a,{{\rm{A}}_1},{{\rm{A}}_2},{{\rm{A}}_3},......,b$ form an A.P.
• If a constant is added to each term of an A.P., then the resulting sequence is also an A.P.
• If a constant is subtracted to each term of an A.P., then the resulting sequence is also an A.P.
• If each term of an A.P. is multiplied by a constant, then the resuting sequence is also an A.P.
• If each term of an A.P. is divided by a constant, then the resuting sequence is also an A.P.
• Sum of first $n$ terms of an A.P.: ${{\rm{S}}_n} = {n \over 2}\left[ {2a + \left( {n - 1} \right)d} \right]$ and ${{\rm{S}}_n} = {n \over 2}\left[ {a + l} \right]$, where $l$ is the last term, i.e., $a+(n-1)d$.

GEOMETRIC PROGRESSION

• A sequence of non-zero numbers is said to be a geometric progression, if the ratio of each term, except the first one, by its preceding term is always the same. $a,ar,a{r^2},.......,a{r^{n - 1}},...$, where $a$ is the first term and $r$ is the common ratio.
• ${n^{th}}$ term of a G.P.: ${a_n} = a{r^{n - 1}}$
• Sum of ${n^{th}}$ terms of a G.P.:  ${{\rm{S}}_n} = {{a\left( {1 - {r^n}} \right)} \over {1 - r}}$if $r<1$.
• Sum to infinity of a G.P.: ${{\rm{S}}_\infty } = {a \over {1 - r}}$
• Geometric mean between a and b: $\sqrt {ab}$
• $n$ Geometric means between a and b: $a.{\left( {{b \over a}} \right)^{{1 \over {n + 1}}}},a.{\left( {{b \over a}} \right)^{{2 \over {n + 1}}}},....a.{\left( {{b \over a}} \right)^{{n \over {n + 1}}}}$
• If all the terms of a G.P. be multiplied or divided by the same quantity the resulting sequence is also a G.P.
• The reciprocal of the terms of a given G.P. form a G.P.
• If each term of a G.P. be raised to the same power, the resulting sequence is also a G.P.

ARITHMETIC – GEOMETRIC SERIES

• A sequence of non-zero numbers is said to be a arithmetic-geometric series, if its terms are obtained on multiplying the terms of an A.P. by the corresponding terms of a G.P. For example:   $1 + {2 \over 3} + {3 \over {{3^2}}} + {4 \over {{3^3}}} + {5 \over {{3^4}}} + .........$
• The general form of an arithmetic-geometric series:
$a,\left( {a + d} \right)r,\left( {a + 2d} \right){r^2},.......,\left[ {a + \left( {n - 1} \right)d} \right]{r^{n - 1}}$
• nth term of an arithmetic-geometric series: ${a_n}$ of A.P. x ${a_n}$ of G.P.
• Sum of n terms of some special series : $\sum n = {{n\left( {n + 1} \right)} \over 2}$
• Sum of squares of first n natural numbers =
• Sum of cubes of fist n natural numebrs = $\sum {{n^3}} = {\left( {{{n\left( {n + 1} \right)} \over 2}} \right)^2} = {\left( {\sum n } \right)^2}$

## The Sequences And Series class 11 Notes

• CBSE Revision notes for Class 11 Mathematics PDF
• CBSE Revision notes Class 11 Mathematics – CBSE
• CBSE Revisions notes and Key Points Class 11 Mathematics
• Summary of the NCERT books all chapters in Mathematics class 11
• Short notes for CBSE class 11th Mathematics
• Key notes and chapter summary of Mathematics class 11
• Quick revision notes for CBSE exams

## CBSE Class-11 Revision Notes and Key Points

Sequences And Series class 11 Notes Mathematics. CBSE quick revision note for class-11 Mathematics, Physics, Chemistry, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.

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