Sequence And Series
An Arithmetic Progression (A.P.):
PYQ2023Sequence_and_SeriesQ4, PYQ2023Sequence_and_SeriesQ11, PYQ2023Sequence_and_SeriesQ13, PYQ2023Sequence_and_SeriesQ15, PYQ2023Sequence_and_SeriesQ16, PYQ2023Sequence_and_SeriesQ19, PYQ2023Sequence_and_SeriesQ22, PYQ2023StatisticsQ1, PYQ2023StatisticsQ4, PYQ2023StatisticsQ10
 $ a, a+d, a+2 d, \ldots \ldots . . a+(n1) d$ is an A.P. . let $a$ be the first term and $d$ be the common difference of an A.P., then
$$ n^{\text {th }} \text{term} =t_{n}=a+(n1) d$$
 $ \ \text{The sum of first} $ $\mathbf{n}$ terms of are A.P.
$$ \mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2 \mathrm{a}+(\mathrm{n}1) \mathrm{d}]=\frac{\mathrm{n}}{2}[\mathrm{a}+\ell] $$
 $ r^{\text {th }}$ term of an A.P. when sum of first $r$ terms is given is
$$t_{r}=S_{r}S_{r1}$$
Properties of A.P.
PYQ2023Sequence_and_SeriesQ1, PYQ2023Sequence_and_SeriesQ12, PYQ2023Binomial_TheoremQ23

If $a, b, c$ are in A.P. $\Rightarrow 2b = a + c$

If $a, b, c, d$ are in A.P. $\Rightarrow a + d = b + c$

Three numbers in A.P. can be taken as $a  d, a, a + d$

Four numbers in A.P. can be taken as $a  3d, a  d, a + d, a + 3d$

Five numbers in A.P. are $a  2d, a  d, a, a + d, a + 2d$

Six terms in A.P. are $a  5d, a  3d, a  d, a + d, a + 3d, a + 5d$ etc.

Sum of the terms of an A.P. equidistant from the beginning and end equals the sum of the first and last term.
Arithmetic Mean (Mean or Average) (A.M.):

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

nArithmetic Means Between Two Numbers:
 If $a, b$ are any two given numbers & $ a, A_{1}, A_{2}, \ldots, A_{n}, b$ are in $A . P$. then $A_{1}, A_{2}, \ldots A_{n}$ are the n A.M.’s between $a$ & $b .$
$$A_{1}=a+\frac{ba}{n+1}, A_{2}=a+\frac{2(ba)}{n+1}, \ldots \ldots, A_{n}=a+\frac{n(ba)}{n+1}$$
$$\sum_{r=1}^{n} A_{r}= nA $$
where A is the single A.M. between $ a $ & $b $
Geometric Progression:
PYQ2023ProbabilityQ5, PYQ2023Sequence_and_SeriesQ2, PYQ2023Sequence_and_SeriesQ6, PYQ2023Sequence_and_SeriesQ7, PYQ2023Binomial_TheoremQ13, PYQ2023ITFQ5, PYQ2023Definite_IntegrationQ13, PYQ2023FunctionQ2, PYQ2023FunctionQ12, PYQ2023MODQ2
$\quad$ a, $a r, a r^{2}, a r^{3}, a r^{4}, \ldots \ldots$ is a G.P. with a as the first term & $r$ as common ratio.

$ n^{\text {th }}$ term $=\operatorname{ar}^{n1}$

Sum of the first $n$ terms i.e. $$ S_{n} = \frac{a(r^n 1)}{r1} , r \ne 1 $$
$$ S_{n} = na , r = 1 $$
 Sum of an infinite G.P. when r < 1 is given by
$$ S_\infty = \frac{a}{1r} , \ \ (r < 1)$$
Geometric Means (Mean Proportional) (G.M.):
PYQ2023Sequence_and_SeriesQ11

If $a, b, c>0$ are in G.P., $b$ is the G.M. between $a$ & $c$, then $b^{2}=a c$

nGeometric Means Between positive number $a$, $b$
If $a, b$ are two given numbers & $a, G_{1}, G_{2}, \ldots . ., G_{n}$, $b$ are in (G.P.).Then $G_{1}, G_{2}, G_{3}, \ldots, G_{n}$ are $n$ G.M.s between $a$ & $b$.
$$G_{1}=a(\frac{b}{a})^{\frac{1}{n+1}}, G_{2}=a(\frac{b}{a})^{\frac{2}{n+1}}, \ldots \ldots, G_{n}=a(\frac{b}{a})^{\frac{n}{n+1}}$$
HARMONICAL PROGRESSION (H.P.):
PYQ2023Sequence_and_SeriesQ9

Harmonical progression is defined as a series in which reciprocal of its terms are in A.P. The standard from of a H.P. is $$ \frac{1}{a}+\frac{1}{a+d}+\frac{1}{a+2 d}+\ldots $$

$a, b, c$ are in H.P. $\Leftrightarrow b=\frac{2 a c}{a+c}$

General term ( $n^{\text {th }}$ term) of a H.P. is given by $T_n=\frac{1}{a+(n1) d}$
Note:

There is no formula and procedure for finding the sum of H.P.

If $a, b, c$ are in H.P. then $\frac{a}{c}=\frac{ab}{bc}$

If $a, b$ are first two terms of an H.P. then $ t_n=\frac{1}{\frac{1}{a}+(n1)\left(\frac{1}{b}\frac{1}{a}\right)}$
HARMONICAL MEAN (H.M.):
$\quad$ If three or more than three terms are in H.P., then all the numbers lying between first and last term are called Harmonical Means between them.
$\quad$ i.e; The H.M. between two given quantities $a$ and $b$ is $\mathrm{H}$ so that $a, H, b$, are in H.P.
$\quad$ i.e. $\frac{1}{\mathrm{a}}, \frac{1}{\mathrm{H}}, \frac{1}{\mathrm{~b}}$ are in A.P. $$ \frac{1}{H}\frac{1}{a}=\frac{1}{b}\frac{1}{H} \Rightarrow H=\frac{2 a b}{a+b} $$
$\quad$ Also $H=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\ldots .+\frac{1}{a_n}}=\frac{n}{\sum_{j=1}^n \frac{1}{a_j}}$
$\quad$ The harmonic mean of $n$ non zero numbers $a_1, a_2, a_3, \ldots \ldots \ldots, a_n$.
Important Results:
PYQ2023Sequence_and_SeriesQ3, PYQ2023Definite_IntegrationQ5, PYQ2023Definite_IntegrationQ6

$$\sum_{r=1}^{n}\left(a_{r} \pm b_{r}\right)=\sum_{r=1}^{n} a_{r} \pm \sum_{r=1}^{n} b_{r}$$

$$\sum_{r=1}^{n} k a_{r}=k \sum_{r=1}^{n} a_{r}$$

$$\sum_{r=1}^{n} k=n k \ , \text{where k is a constant} $$

Sum of first $n$ natural numbers
$$\sum_{r=1}^{n} r=1+2+3+\ldots \ldots \ldots . .+n=\frac{n(n+1)}{2}$$

Sum of first $\mathrm{n}$ odd natural numbers $$ \Rightarrow \sum_{r=1}^n(2 r1)=n^2 $$

Sum of first $n$ even natural numbers $$ \Rightarrow \sum_{r=1}^n 2 r=n(n+1) $$

Sum of squares of first $n$ natural numbers
$$\sum_{r=1}^{n} r^{2}=1^{2}+2^{2}+3^{2}+\ldots \ldots \ldots \ldots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$$
 Sum of cubes of first $n$ natural numbers
$$ \sum_{r=1}^{n} r^{3}=1^{3}+2^{3}+3^{3}+\ldots \ldots \ldots . .+n^{3}=\frac{n^{2}(n+1)^{2}}{4} $$

$$ 2 \sum_{i<j=1}^{n} a_{i} a_{j}=(a_{1}+a_{2}+\ldots \ldots \ldots+a_{n})^{2}(a_{1}^{2}+a_{2}^{2}+\ldots \ldots+a_{n}^{2}) $$

Sum of fourth powers of first $n$ natural numbers $\left(\sum n^4\right)$ $$ \sum n^4=1^4+2^4+\ldots . .+n^4=\frac{n(n+1)(2 n+1)\left(3 n^2+3 n1\right)}{30} $$

If $r^{\text {th }}$ term of an A.P. $$ T_r=A r^3+B r^2+C r+D \text {, then } $$ $\quad \quad \quad$ sum of $n$ term of AP is $$ S_n=\sum_{r=1}^n T_r=A \sum_{r=1}^n r^3+B \sum_{r=1}^n r^2+C \sum_{r=1}^n r+D \sum_{r=1}^n 1 $$

Relation between means
$\quad \quad \mathrm{G}^{2}=\mathrm{AH}, \quad$ A.M. $\geq$ G.M. $\geq$ H.M. and A.M. $=$ G.M. $=$ H.M.
$\quad \quad $ if $a_{1}=a_{2}=a_{3}=\ldots \ldots \ldots . .=a_{n}$
Important Note:
PYQ2023Sequence_and_SeriesQ5, PYQ2023StatisticsQ10

If for an A.P. $p^{\text {th }}$ term is $q, q^{\text {th }}$ term is $p$ then $m^{\text {th }}$ term is $=p+qm$

If for an AP sum of $p$ terms is $q$, sum of $q$ terms is $p$, then sum of $(p+q)$ term is $(p+q)$.

If for an A.P. sum of $p$ terms is equal to sum of $q$ terms then sum of $(p+q)$ terms is zero

If sum of $n$ terms $S_n$ is given then general term $T_n=S_nS_{n1}$ where $S_{n1}$ is sum of $(n1)$ terms of A.P.

Common difference of AP is given by $d=S_22 S_1$ where $S_2$ is sum of first two terms and $\mathrm{S}_1$ is sum of first term or first term.

The sum of infinite terms of an A.P. is $\infty$ if $d>0$ and $\infty$ if $d<0$

Sum of $n$ 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. $2 A$

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

If for the different A.P.’s $\frac{S_n}{S_n^{\prime}}=\frac{f_n}{\phi_n}$ then $\frac{T_n}{T_n^{\prime}}=\frac{f(2 n1)}{\phi(2 n1)}$

If for two A.P.’s $\frac{T_n}{T_n^{\prime}}=\frac{A n+B}{C n+D}$
ArithmeticoGeometrical Progression (A.G.P.):
PYQ2023Sequence_and_SeriesQ8, PYQ2023Sequence_and_SeriesQ10, PYQ2023Sequence_and_SeriesQ17, PYQ2023Sequence_and_SeriesQ18
$\quad$ If each term of a progression is the product of the corresponding terms of an A.P. and a G.P., then it is called arithmeticgeometric progression (A.G.P.)
e.g. $$a,(a+d) r,(a+2 d) r^2, \ldots \ldots$$.

The general term ( $n^{\text {th }}$ term) of an A.G.P. is $$ T_n=[a+(n1) d] r^{n1} $$

To find the sum of $n$ terms of an A.G.P. we suppose its sum $S$, multiply both sides by the common ratio of the corresponding G.P. and then subtract as in following way and
we get a G.P. whose sum can be easily obtained. $$ \begin{aligned} & S_n=a+(a+d) r+(a+2 d) r^2+\ldots . .[a+(n1) d] r^{n1} \ & r S_n=\quad a r+(a+d) r^2+\ldots .+[a+(n1) d] r^n \end{aligned} $$
$\quad$ After subtraction we get $$ S_n(1r)=a+r . d+r^2 \cdot d \ldots . . d r^{n1}[a+(n1) d] r^n $$
$\quad$ After solving $$ S_n=\frac{a}{1r}+\frac{r \cdot d\left(1r^{n1}\right)}{(1r)^2} $$ $\quad$ and $$S_{\infty}=\frac{a}{1r}+\frac{d r}{(1r)^2}$$
$\quad $ Note : This is not a standard formula. This is only to understand the procedure for finding the sum of an A.G.P. However formula for sum of infinite terms can be used directly.