Arithmetic, Geometric And Harmonic Progressions

Arithmetic Progression (AP)

A sequence in which the difference of two consecutive terms is constant, is called Arithmetic progression (AP)

$a,a+d,a+2d$ is an AP, where $a,=$first term and $d=$ common difference.

$nth$ Term of an AP If $a$is the first term, $d$ is the common difference and $l$ is the last term of an AP, i.e.

$a,a+d,a+2d,a+3d,...l$

1 $n$th term is given by $l$$=a_n=a+\left(n-1\right)d$ 2 $n$th term of an AP from the last term is ${aapos}_n=l-\left(n-1\right)d$ 3 $a_n+{aapos}_n=a+l$i.e. $n$th term from the start +$n$th term from the end = first term +last term4. Common difference of an AP $d=T_n-T_{n-1,}\forall n>1$5. ${T_n} = \frac{1}{2}\left[ {{T_{n - k}} + {T_{n + k}}} \right],k < n$Properties of Arithmetic Progression 1 Ifa constant is added or subtracted from each term of an AP then the resulting sequence is an AP with same common difference. 2 If each term of an AP is multiplied or divided by a non-zero constant $k,$ then theresulting sequence is also an AP, with common difference $kd$or $\frac{d}{k},$ where $d=$ common difference. 3 If $a_n,a_{n+1}$and $a_{n+2}$ are three consecutive terms of an AP, then $2a_{n+1}=a_n+a_{n+2.}$ 4 If the terms of an AP are chosen at regular intervals, then they form an A.P,Selection of Terms in an AP 1 Any three termsof AP can be taken of AP can be taken as$\left(a-d\right),a,\left(a+b\right)$ 2 Any four terms of AP can be taken as$\left(a-3d\right),\left(a-d\right),\left(a+d\right),\left(a+3d\right)$ 3 Any five terms of AP can be taken as$\left(a-2d\right),\left(a-d\right),a,\left(a+d\right),\left(a+2d\right)$Sum of $n$ terms of an AP • Sum of $n$ terms of AP, is given by$S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]=\frac{n}{2}\left[a+l\right],$ where$l=$last term• A sequence is an AP, iff the sum of $n$ terms is of theform $A{n}^2+Bn,$where $A$and $B$are constants and common difference in such case will be $2A.$• $T_n=S_n-S_{n-1}i.e.$ $n$th term of AP=Sum of $n$terms-Sum of $\left(n-1\right)$ termsArithmetic Mean (AM) 1 If $a,A,b$ are inAP, then $A=\frac{a+b}{2}$ is called the arithmetic mean of $a$and $b$. 2 If$a_1,a_2,a_3,....a_n$are $n$ numbers, then their AM is given by, $A=\frac{a_1+a_2+.....+a_n}{n}$ 3 If$a,A_1,A_2,A_3,......A_n,b$ are in AP, then$A_1,A_2,A_3,......A_n$ are $n$ arithmetic mean between $a$ and $b,$ where$d=\frac{b-a}{n+1}$$A_1=a+d=\frac{na+b}{n+1}$$A_2=a+2d=\frac{\left(n+1\right)a+2b}{n+1}$...$A_n=a+nd=\frac{a+nd}{n+1}$Geometric Progression GPA sequence in which the ratio of two consecutive terms isconstant called GP. The constant ratio is common ratio $\left(r\right).$i.e. $\frac{a_{n+1}}{a_n}=r,\forall n\ge 1$If $a$ is the first term and $r$ is the common ratio of a G.P, then the GP can be written as $a,ar,a{r}^2,........a{r}^{n-1},.......$$n$th Term of a GP If $a$is the first term and $r$ is the common ratio 1 $n$th term of a GP from the beginning is $a_n=a{r}^{n-1}$ 2 $n$thterm of a GP from the end is $a_n^{apos}=\frac{l}{r^{n-1}},$ where $l=$ last term $a{r}^{m-n},$ where $a$ is the first term and $r$ is the common ratio of the GP. 3 $a_n{aapos}_n=al$i.e. $n$th term from the beginning $\times$ $n$th term from the end=first term$\times$last termProperties of Geometric Progression 1 If all the terms of GP be multiplied or divided by same non-zero constant, then the resulting sequence is a GP with the same common ratio. 2 The reciprocal terms of a given GP form a GP. 3 If each term of a GP be raised to same power, then the resulting sequence also forms a GP. 4 If the terms of a GP are chosen at regular intervals,then the resulting sequence is also a GP. 5 If $a_1,a_2,a_3,......,a_n$ are non-zero and non –negative term of a GP, then$\log a_1,log{a}_2,log{a}_3,......,log{a}_n$ are in an GP, then $b^2=ac.$Selection of Terms in a GP 1 Anythree terms of a GP can be taken as$\frac{a}{r},a$and $ar.$ 2 Any four terms of a GP can be taken as $\frac{a}{r^3},\frac{a}{r},$$ar$and $a{r}^3.$

3 Any five terms of a GP can be taken as $\frac{a}{r^2},\frac{a}{r},$$a$,$ar$ and $a{r}^2.$

Sum of $n$ Terms of a GP

• Sum of $n$ terms of a GP is given by

$S_n=\{\begin{array}{c}\frac{a\left(1-r^n\right)}{1-r},ifr<1\\ \frac{a\left(r^r-1\right)}{r-1},ifr>1\\ na,ifr=1\end{array}$

• $S_n=\frac{a-lr}{1-r},r<1$ or $S_n=\frac{lr-a}{r-1},r>1$ where, $l=$ last term of the GP

• If$\left|r\right|<1,$ then $S_{\infty }=\frac{a}{1-r}$

• If $\left|r\right|\ge 1,$ then it does not exist.

Geometric mean GM

1 If $a,G,b$ are in $GP,$ them $G$ is called the geometric mean of $a$ are and $b$ is given by $G=\sqrt{ab}$

2 If $a,G_1,G_2,G_3,....G_n,b$ are in GP, then $G_1,G_2,G_3,....G_n$are in $G{M}^{\prime }s$ between $a$ and $b,$where

$r={\left(\frac{b}{a}\right)}^{\frac{1}{n+1}},$

$G_1=ar=a{\left(\frac{b}{a}\right)}^{\frac{1}{n+1}}$

$G_2=a{r}^2=a{\left(\frac{b}{a}\right)}^{\frac{2}{n+1}}$

……………………………….

$G_n=a{r}^n=a{\left(\frac{b}{a}\right)}^{\frac{n}{n+1}}$

1 Product of $n$ GM's,

$G_1\times G_2\times G_3\times ....\times G_n=G$

Harmonic Progression (HP)

A Sequence $a_1,a_2,a_3,....a_n$ of non-zero numbers is called a Harmonic Progression (HP), if the sequence $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},.......,\frac{1}{a_n}$ is an AP.

1. $n$th term of the HP from the beginning

$a_n=\frac{1}{\frac{1}{a_n}+\left(n-1\right)\left(\frac{1}{a_2}-\frac{1}{a_1}\right)}$

$=\frac{a_1{a}_2}{a_2+\left(n-1\right)\left(a_1-a_2\right)}$

2. $n$th term of the HP from the end

${a^{\prime }}_n=\frac{1}{\frac{1}{a_n}-\left(n-1\right)\left(\frac{1}{a_2}-\frac{1}{a_1}\right)}$ $=\frac{a_1{a}_2{a}_n}{a_1{a}_2-a_n\left(n-1\right)\left(a_1-a_2\right)}$

3. $\frac{1}{a_n}+\frac{1}{{a^{\prime }}_n}=\frac{1}{a}+\frac{1}{a_n}$

$=\frac{1}{firsttermofAP}+\frac{1}{lasttermofAP}$

4. $a_n=\frac{1}{a+\left(n-1\right)d},$if $a,d$ are the first term and common difference of the corresponding AP.

Note There is no formula for determining the sum of harmonic series.

Harmonic Mean

1 If $a,H,b$ are in $HP,$ then $H$ is called the harmonic mean of $a$ and $b$ and is given by $H=\frac{2ab}{a+b}$

2 If $a,H_1,H_2,H_3......H_n,b$ are in HP, then $H_1,H_2,H_3......H_n$ are $n$ harmonic means between $a$and $b$, where

$H_1=\frac{\left(n+1\right)ab}{a+nb}$

$H_2=\frac{\left(n+1\right)ab}{2a+\left(n-1\right)b},.........$

3. Harmonic Mean $\left(HM\right)$ between $a_1,a_2,a_3......a_n$is given by

$\frac{1}{H}=\frac{1}{n}\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+.......+\frac{1}{a_n}\right)$

Properties of AM,GM and HM between Two Numbers

If $A,G$and $H$ are arithmetic, geometric and harmonic means of two positive numbers $a$ and $b,$ then

1. $A=\frac{a+b}{2},G=\sqrt{ab},H=\frac{2ab}{a+b}$

2. $A\ge G\ge H$

3. If $A,G,H$ are in $GP,$ then $G^2=AH$

4. If $A,G,H$and $AM,GM,$ and $HM$ between three given numbers $a,b$ and $c$ then the equation on having $a,b$ and $c$ as its root is

$x^3-3A{x}^2+\frac{3G^3}{H}x-G^3=0,$

Where $A=\frac{a+b+c}{3},G={\left(abc\right)}^{1/3}$

And $\frac{1}{H}=\left(\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3}\right)$

5. If $A_1,A_2$ be two $A{M}^{\prime }s,G_1,G_2$ be two $G{M}^{\prime }s$ and $H_1,H_2$ be two $H{M}^{\prime }s$ between two numbers $a$ and $b$ then

$\frac{G_1{G}_2}{H_1{H}_2}=\frac{A_1+A_2}{H_1+H_2}$

6. If $A,G$ and $H$ be $AM,GM$ and $HM$ between two numbers $a$ and $b,$ then

$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\{\begin{array}{c}Aifn=0\\ Gifn=-\frac{1}{2}\\ Hifn=-1\end{array}$