SHORT NOTES: Sequences and Series

If $a_1,a_2,a_3,...,a_n$ is a sequence, then the expression $a_1+a_2+a_3+...+a_n$ is a series.
Those sequences whose terms follow certain patterns are called progressions.

Arithmetic Progression(A.P):

$a_{n+1}-a_n=constant(=d)$ for all $n\in N$

If $a$ is the first term and $d$ the common difference of an A.P.,then its $nth$ term is

$a_n=a+(n-1)d$

Sum of 'n' terms of an A.P. :

$S_n=\frac{n}{2}[2a+(n-1)d]$

$S_n=\frac{n}{2}[a+l]$ where l = last term = $a+(n-1)d$

Geometric Progression(G.P.) :

$\frac{a_{n+1}}{a_n}=r=constant$ for all $n\in N$

If $a$ is the first term and $r$ is common ratio then its $nth$ term is

$a_n=ar^{n-1}$

Sum of 'n' terms of an G.P. :

$S_n=a\frac{(1-r^n)}{1-r}$ For $\left | r \right |< 1$

$S_n=a\frac{(r^n-1)}{r-1}$ For $\left | r \right |> 1$

Infinite Geometric Series Sum

$S=\lim_{n\rightarrow \infty }S_n=\frac{a}{1-r}$ For $\left | r \right |< 1$ , $n\rightarrow \infty$

harmonic progressions(H.P.) :

$a_1,a_2,a_3,....,a_n$ is a Harmonic Progression if the sequence $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},....,\frac{1}{a_n}$ is an A,P,

arithmetic(A), geometric(G) and harmonic means(H) :

Two positive numbers a and b

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

Similarly for n arithmetic / Geometric mean between a and b find d / r as

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

So nth arithmetic / geometric mean is

$A_n=a+nd,G_n=ar^n$

Notes:

$A\geq G\geq H$
$G^2=AH$
$x^2-2Ax+G^2=0$ have 'a' and 'b' as roots.

Arithmetic-Geometric Progression(AGP) :

$a,(a+d){\color{Blue} r},(a+2d){\color{Blue} r^2},.....$

Sum of n terms of AGP :

$S_n=\frac{a}{1-r}+dr\frac{(1-r^{n-1})}{(1-r)^2}-\frac{\left \{ a+(n-1)d \right \}r^n}{1-r}$

Sum of infinite AGP :

$S_n=\frac{a}{1-r}+\frac{dr}{(1-r)^2}$

Notes:

$\sum_{r=1}^{n}r=1+2+3+...+n=\frac{n\left ( n+1 \right )}{2}$

$\sum_{r=1}^{n}r^2=1^2+2^2+3^2+...+n^2=\frac{n\left ( n+1 \right )\left ( 2n+1 \right )}{6}$

$\sum_{r=1}^{n}r^3=1^3+2^3+3^3+...+n^3=\left \{ \frac{n\left ( n+1 \right )}{2} \right \}^2$

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Sequences and Series

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