Sequences and Series - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes

Chapter-9

SEQUENCES AND SERIES

---

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 a1, a2, a3${\text{a}}_1,{\text{a}}_2,{\text{a}}_3$, ... be the sequence, then the sum expressed as a1+ a2+ a3+.....${\text{a}}_1\text{+ }{\text{a}}_2\text{+ }{\text{a}}_3+.....$  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$l$. The general term or the  nth  term of the A.P. is given by an= a + (n − 1) d.${\text{a}}_n=\text{ a }+\left(\text{n }-\text{ 1}\right)\text{ d}.$
• Single Arithmetic mean between any two given numbers a and b: A.M. = a+b2$\frac{a+b}{2}$
• n$n$ Arithmetic mean between two given numbers a and b: a,A1,A2,A3,......,b$a,{\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3,......,b$ 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$n$ terms of an A.P.: Sn=n2[2a+(n−1)d]${\mathrm{S}}_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$ and Sn=n2[a+l]${\mathrm{S}}_n=\frac{n}{2}\left[a+l\right]$, where l$l$ is the last term, i.e., a+(n−1)d$a+(n-1)d$.

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,ar2,.......,arn−1,...$a,ar,a{r}^2,.......,a{r}^{n-1},...$, where a$a$ is the first term and r$r$ is the common ratio.
• nth$n^{th}$ term of a G.P.: an=arn−1$a_n=a{r}^{n-1}$
• Sum of nth$n^{th}$ terms of a G.P.:  Sn=a(1−rn)1−r${\mathrm{S}}_n=\frac{a\left(1-r^n\right)}{1-r}$if r<1$r<1$.
• Sum to infinity of a G.P.: S∞=a1−r${\mathrm{S}}_{\mathrm{\infty }}=\frac{a}{1-r}$
• Geomtric mean between a and b: ab−−√$\sqrt{ab}$
• n$n$ Geometric means between a and b: a.(ba)1n+1,a.(ba)2n+1,....a.(ba)nn+1$a.{\left(\frac{b}{a}\right)}^{\frac{1}{n+1}},a.{\left(\frac{b}{a}\right)}^{\frac{2}{n+1}},....a.{\left(\frac{b}{a}\right)}^{\frac{n}{n+1}}$
• 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+23+332+433+534+.........$1+\frac{2}{3}+\frac{3}{3^2}+\frac{4}{3^3}+\frac{5}{3^4}+.........$
• The general form of an arithmetic-geometric series:
  a,(a+d)r,(a+2d)r2,.......,[a+(n−1)d]rn−1$a,\left(a+d\right)r,\left(a+2d\right)r^2,.......,\left[a+\left(n-1\right)d\right]r^{n-1}$
• nth term of an arithmetic-geometric series: an$a_n$ of A.P. x an$a_n$ of G.P.
• Sum of n terms of some special series : ∑n=n(n+1)2$\sum n=\frac{n\left(n+1\right)}{2}$
• Sum of squares of first n natural numbers = ∑n2=n(n+1)(2n+1)6$\sum n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$
• Sum of cubes of fist n natural numebrs = ∑n3=(n(n+1)2)2=(∑n)2