Arithmetic Progression (A.P.):

• $ a, a+d, a+2 d, \ldots \ldots . . a+(n-1) 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+(n-1) 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

$t_{r}=S_{r}-S_{r-1}$

Properties of A.P.

• 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.

• n-Arithmetic 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 $a$ & $b .$

  $A_{1}=a+\frac{b-a}{n+1}, A_{2}=a+\frac{2(b-a)}{n+1}, \ldots \ldots, A_{n}=a+\frac{n(b-a)}{n+1}$

  $\sum_{r=1}^{n} A_{r}= nA $

  where A is the single A.M. between $ a $ & $b $

Geometric Progression:

$\quad$ 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 $=\operatorname{ar}^{n-1}$

• Sum of the first $n$ terms i.e. $ S_{n} = \frac{a(r^n -1)}{r-1} , r \ne 1 $

$ S_{n} = na , r = 1 $

• Sum of an infinite G.P. when |r| < 1 is given by

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

Geometric Means (Mean Proportional) (G.M.):

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

• n-Geometric 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 $a$ & $b$.

$G_{1}=a(\frac{b}{a})^{\frac{1}{n+1}}, G_{2}=a(\frac{b}{a})^{\frac{2}{n+1}}, \ldots \ldots, G_{n}=a(\frac{b}{a})^{\frac{n}{n+1}}$

HARMONIC PROGRESSION (H.P.):

• Harmonic progression is defined as a series in which reciprocal of its terms are in A.P. If $a_1,a_2, \ldots $ are in HP then $ \frac{1}{a_1},\frac{1}{a_2}, \ldots $ is in AP < !– 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. $\Leftrightarrow b=\frac{2 a c}{a+c} = \frac{a}{c}=\frac{a-b}{b-c}$

• For HP whose first term is $a$ and second term is $b$, the $n^{th}$ term is $t_n=\frac{ab}{b+(n-1)(a-b)}$

Note:

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

HARMONIC 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 Harmonic 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:

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

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

• $\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 r-1)=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} $

• 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 n-1\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. if $a_{1}=a_{2}=a_{3}=\ldots \ldots \ldots . .=a_{n}$

Important Note:

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

• 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 $T_n=S_n-S_{n-1}$ where $S_{n-1}$ is sum of $(n-1)$ terms of A.P.

• Common difference of AP is given by $d=S_2-2 S_1$ 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 $\frac{T_n}{T_n^{\prime}}=\frac{f(2 n-1)}{\phi(2 n-1)}$

• If for two A.P.’s $\frac{T_n}{T_n^{\prime}}=\frac{A n+B}{C n+D}$

Arithmetic-Geometric Progression (A.G.P.):

$\quad$ 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 arithmetic-geometric 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+(n-1) d] r^{n-1} $

• 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+(n-1) d] r^{n-1} \ & r S_n=\quad a r+(a+d) r^2+\ldots .+[a+(n-1) d] r^n \end{aligned} $

  $\quad$ After subtraction we get $ S_n(1-r)=a+r . d+r^2 \cdot d \ldots . . d r^{n-1}-[a+(n-1) d] r^n $

  $\quad$ After solving $ S_n = \frac{a (1 - r^n)}{1 - r} + \frac{dr (1 - r^{n-1})}{(1 - r)^2} $ $\quad$ and $S_{\infty}=\frac{a}{1-r}+\frac{d r}{(1-r)^2}$

  $\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.