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Sequence And Series - Notes, Concept and All Important Formula

SEQUENCE & SERIES

1. ARITHMETIC PROGRESSION (AP) :

AP is sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference. If ‘a’ is the first term & ‘d’ is the common difference, then AP can be written as a, a + d, a + 2d, ..., a + (n – 1) d, ...

(a) \(n^{\text {th }}\) term of this AP \(\boxed{T_{n}=a+(n-1) d}\), where \(d=T_{n}-T_{n-1}\)

(b) The sum of the first \(n\) terms : \(\boxed{S_{n}=\frac{n}{2}[2 a+(n-1) d]=\frac{n}{2}[a+\ell]}\),  where \(\ell\) is the last term.

(c) Also \(n ^{\text {th }}\) term \(\boxed{T _{ n }= S _{ n }- S _{ n -1}}\)


Note:

(i) Sum of first n terms of an A.P. is of the form \(A n^{2}+B n\) i.e. a quadratic expression in n, in such case the common difference is twice the coefficient of \(n ^{2}\). i.e. 2A

(ii) \(n ^{\text {th }}\) term of an A.P. is of the form \(An + B\) i.e. a linear expression in \(n\), in such case the coefficient of \(n\) is the common difference of the A.P. i.e. \(A\)

(iii) Three numbers in AP can be taken as a – d, a, a + d; four numbers in AP can be taken as a – 3d, a – d, a + d, a + 3d five numbers in AP are a – 2d, a – d, a, a + d, a + 2d & six terms in AP are a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d etc.

(iv) If a, b, c are in A.P., then \(b=\frac{a+c}{2}\)

(v) If \(a_{1}, a_{2}, a_{3} \ldots \ldots\) and \(b_{1}, b_{2}, b_{3} \ldots \ldots \ldots\) are two A.P.s, then \(a_{1} \pm b_{1}, a_{2} \pm b_{2}, a_{3} \pm b_{3} \ldots \ldots \ldots\) are also in A.P.

(vi) (a) If each term of an A.P. is increased or decreased by the same number, then the resulting sequence is also an A.P. having the same common difference.

(b) If each term of an A.P. is multiplied or divided by the same non zero number (k), then the resulting sequence is also an A.P. whose common difference is \(kd\)  &  \(d / k\) respectively, where d is common difference of original A.P.

(vii) Any term of an AP (except the first & last) is equal to half the sum of terms which are equidistant from it.

\(T _{ r }=\frac{ T _{ r - k }+ T _{ r + k }}{2}, k < r\)


2. GEOMETRIC PROGRESSION (GP) :

GP is a sequence of numbers whose first term is non-zero & each of the succeeding terms is equal to the preceding terms multiplied by a constant. Thus in a GP the ratio of successive terms is constant. This constant factor is called the common ratio of the series & is obtained by dividing any term by the immediately previous term. Therefore a, ar, \(a r^{2}, a r^{3}, a r^{4}, \ldots \ldots \ldots .\) is a GP with 'a' as the first term & 'r' as common ratio.

(a) \(n ^{\text {th }}\) term \(T _{ n }= ar ^{ n -1}\)

(b) Sum of the first \(n\) terms \(S _{ n }=\frac{ a \left( r ^{ n }-1\right)}{ r -1}\), if \(r \neq 1\)

(c) Sum of infinite GP when \(|r|<1 \quad\left(n \rightarrow \infty, r^{n} \rightarrow 0\right)\)

\(S_{\infty}=\frac{a}{1-r} ;|r|<1\)

(d) If \(a , b , c\) are in \(GP \Rightarrow b ^{2}= ac \Rightarrow\) loga, logb, logc, are in A.P.


Note:

(i) In a G.P. product of \(k ^{\text {th }}\) term from beginning and \(k ^{\text {th }}\) term from the last is always constant which equal to product of first term and last term.

(ii) Three numbers in G.P. : \(\mathbf{\quad a / r , a , a r}\)

Five numbers in G.P. \(\mathbf{\quad: \quad a / r ^{2}, a / r , a , a r , a r ^{2}}\)

Four numbers in G.P. \(\mathbf{ : \quad a / r ^{3}, a / r}\), ar, ar \(\mathbf{ ^{3}}\)

Six numbers in G.P. \(\mathbf{ \quad: \quad a / r ^{5}, a / r ^{3}, a / r}\), ar, \(\mathbf{ a r ^{3}, a r ^{5}}\)

(iii) If each term of a G.P. be raised to the same power, then resulting series is also a G.P.

(iv) If each term of a G.P. be multiplied or divided by the same non-zero quantity, then the resulting sequence is also a G.P.

(v) If \(a_{1}, a_{2}, a_{3} \ldots .\) and \(b_{1}, b_{2}, b_{3}, \ldots \ldots\) be two G.P.'s of common ratio \(r _{1}\) and \(r _{2}\) respectively, then \(a _{1} b _{1}, a _{2} b _{2} \ldots \ldots\) and \(\frac{ a _{1}}{ b _{1}}, \frac{ a _{2}}{ b _{2}}, \frac{ a _{3}}{ b _{3}} \ldots \ldots\) will also form a G.P. common ratio will be \(r _{1} r _{2}\) and \(\frac{ r _{1}}{ r _{2}}\) respectively

(vi) In a positive G.P. every term (except first) is equal to square root of product of its two terms which are equidistant from it. i.e. \(\mathbf{ T _{ r }=\sqrt{ T _{ r - k } T _{ r + k }}, k < r}\)

(vii) If \(\mathbf{ a _{1}, a _{2}, a _{3} \ldots . a _{ n }}\) is a. G.P. of non zero, non negative terms, then \(\mathbf{ \log a _{1}, \log a _{2}, \ldots . . \log a _{ n }}\) is an A.P. and vice-versa.


3. HARMONIC PROGRESSION (HP) :

A sequence is said to \(HP\) if the reciprocals of its terms are in AP. If the sequence \(a _{1}, a _{2},  a _{3}, \ldots, a _{ n }\) is an HP then \(1 / a _{1}, 1 / a _{2}, \ldots .\)  \(1 / a _{ n }\) is an AP & vice-versa. Here we do not have the formula for the sum of the \(n\) terms of an HP. The general form of a harmonic progression is  \(\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2 d}, \ldots \ldots \cdot \frac{1}{a+(n-1) d}\)

Note : No term of any H.P. can be zero. If \(a , b , c\) are in HP  \(\Rightarrow b =\frac{2 ac }{ a + c } \text { or } \frac{ a }{ c }=\frac{ a - b }{ b - c }\)


4. MEANS

(a) Arithmetic mean (AM):

If three terms are in AP then the middle term is called the AM between the other two, so if \(a, b, c\) are in \(A P, b\) is \(A M\) of \(a \)  &   \(c\).

n-arithmetic means between two numbers:

If \(a, b\) are any two given numbers & \( a, A_{1}, A_{2}, \ldots \ldots ., A_{n}, b\) are in AP then \(A_{1}, A_{2}, \ldots A_{n}\) are the \(n\) AM's between \(a \)  &   \(b\), then \(A_{1}=a+d, A_{2}=a+2 d, \ldots \ldots, A_{n}=a+n d\), where \(d=\frac{b-a}{n+1}\)

Note : Sum of n AM's inserted between a & b is equal to \(n\) times the single AM between a & b i.e. \(\displaystyle \sum _{r=1}^{n} A_{r}=n A\) where \(A\) is the single AM between a & b i.e. \(\frac{a+b}{2}\)

(b) Geometric mean (GM):

If \(a, b, c\) are in GP, then \(b\) is the GM between a & c i.e. \(b^{2}=a c\) therefore \(b =\sqrt{ ac }\)

n-geometric means between two numbers:

If \(a, b\) are two given positive numbers & \( a, G_{1}, G_{2}, \ldots \ldots . ., G_{n}\) \(b\) are in GP then \(G _{1}, G _{2}, G _{3}, \ldots \ldots G _{ n }\) aren GMs between \(a \& b\). \(G _{1}= ar , G _{2}= ar ^{2}, \ldots \ldots . G _{ n }= ar ^{ n }\), where \(r =( b / a )^{1 / n +1}\)

Note : The product of \(n\) GMs between \(a\)   &  \(b\) is equal to nth power of the single GM between a & bi.e. \(\prod_{r=1}^{n} G_{r}=(G)^{n}\) where G is the single GM between a  & b i.e. \(\sqrt{a b}\)

(c) Harmonic mean (HM) :

If \(a, b, c\) are in \(H P\), then \(b\) is \(H M\) between \(a \)  &   \(c\), then \(b=\frac{2 a c}{a+c}\).

Important note :

(i) If \(A , G , H\), are respectively AM, GM, HM between two positive number a & b then

(a) \(G ^{2}= AH ( A , G , H\) constitute \(a GP )\)  (b) \(A \geq G \geq H\)  (c) \(A=G=H \Leftrightarrow a=b\)

(ii) Let \(a_{1}, a_{2}, \ldots \ldots, a_{n}\) be \(n\) positive real numbers, then we define their arithmetic mean (A), geometric mean (G) and harmonic mean (H) as \(A=\dfrac{ a _{1}+ a _{2}+\ldots .+ a _{ n }}{ n }\)

\(G =\left( a _{1} a _{2} \ldots \ldots \ldots a _{ n }\right)^{1 / n }\) and \(H =\dfrac{ n }{\left(\frac{1}{ a _{1}}+\frac{1}{ a _{2}}+\frac{1}{ a _{3}}+\ldots .+\frac{1}{ a _{ n }}\right)}\)

It can be shown that \(A \geq G \geq H\). Moreover equality holds at either place if and only if \(a_{1}=a_{2}=\ldots \ldots=a_{n}\)


5. ARITHMETICO - GEOMETRIC SERIES :

Sum of First n terms of an Arithmetico-Geometric Series:

Let \(S_{n}=a+(a+d) r+(a+2 d) r^{2}+\ldots \ldots\)\( \ldots+[a+(n-1) d] r^{n-1}\) then  \(S_{n}=\frac{a}{1-r}+\frac{d r\left(1-r^{n-1}\right)}{(1-r)^{2}}-\frac{[a+(n-1) d] r^{n}}{1-r}, r \neq 1\)

Sum to infinity:

If \(|r|<1 \& n \rightarrow \infty\) then \(\displaystyle \lim_{n \rightarrow \infty} r^{n}=0 \Rightarrow S_{\infty}=\frac{a}{1-r}+\frac{d r}{(1-r)^{2}}\)


6. SIGMA NOTATIONS

Theorems:

(a) \(\displaystyle \sum _{ r =1}^{ n }\left( a _{ r } \pm b _{ r }\right)=\displaystyle \sum _{ r =1}^{ n } a _{ r } \pm \displaystyle \sum _{ r =1}^{ n } b _{ r }\)

(b) \(\displaystyle \sum _{ r =1}^{ n } ka _{ r }= k \displaystyle \sum _{ r =1}^{ n } a _{ r }\)

(c) \(\displaystyle \sum _{ r =1}^{ n } k = nk ;\) where \(k\) is a constant.


7. RESULTS

(a) \(\displaystyle \sum _{ r =1}^{ n } r =\frac{ n ( n +1)}{2}\) (sum of the first \(n\) natural numbers)

(b) \(\displaystyle \sum _{ r =1}^{ n } r ^{2}=\frac{ n ( n +1)(2 n +1)}{6}\) (sum of the squares of the first \(n\) natural numbers)

(c) \(\displaystyle \sum _{r=1}^{n} r^{3}=\frac{n^{2}(n+1)^{2}}{4}=\left[\displaystyle \sum _{r=1}^{n} r\right]^{2}\) (sum of the cubes of the first \(n\) natural numbers)

(d) \(\displaystyle \sum _{ r =1}^{ n } r ^{4}=\frac{ n }{30}( n +1)(2 n +1)\left(3 n ^{2}+3 n -1\right)\)

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