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Arithmetic Progression (A.P.)

$\dpi{80} \\\text{For an arithmetic progression (A.P.) whose 1st term is a and } \\\text{the common difference is d.} \\\text{ i ) } n^{th} \text{ term = }t_n = a + (n-1)d \\\text{ ii )The sum of the 1st (n) terms )} S_n = \frac{n}{2}(a+l) \\\text{where l = last term = a + (n-1)d} \\\therefore S_n = \frac{n}{2}\left \{ 2a + (n -1)d \right \}$

$\dpi{80} \\\text{For an arithmetic progression (A.P.) whose 1st term is a and } \\\text{the common difference is d.} \\\text{ i ) } n^{th} \text{ term = }t_n = a + (n-1)d \\\text{ ii )The sum of the 1st (n) terms )} S_n = \frac{n}{2}(a+l) \\\text{where l = last term = a + (n-1)d} \\\therefore S_n = \frac{n}{2}\left \{ 2a + (n -1)d \right \}$

The roots of the quadratic equation

$\dpi{60} \\\text{*If } \alpha \text{ and } \beta \text{ are the roots of the equation } ax^2 + bx + c = 0; a\neq 0 \text{ then} \\\text{ i) } \alpha + \beta = -\frac{b}{a} =- \frac{\text{ coefficient of x}}{\text{coefficient of }x^2}\\\ \\\text{ ii) } \alpha . \beta =\frac{c}{a}= \frac{\text{ coefficient term}}{\text{coefficient of }x^2}\\$

$\dpi{60} \\\text{*If } \alpha \text{ and } \beta \text{ are the roots of the equation } ax^2 + bx + c = 0; a\neq 0 \text{ then} \\\text{ i) } \alpha + \beta = -\frac{b}{a} =- \frac{\text{ coefficient of x}}{\text{coefficient of }x^2} \\\text{ ii) } \alpha . \beta =\frac{c}{a}= \frac{\text{ coefficient term}}{\text{coefficient of }x^2}\\\text{The quadratic equation whose roots are } \alpha \text{ and } \beta \text{ is }(x-\alpha)(x-\alpha)=0 \\\text{ i.e }x^2-(\alpha + \beta)x+\alpha .\beta=0 \\\text{ i.e }x^2-Sx+P=0 \\\text{, where S = Sum of the roots and P = Product of the roots}$

$\dpi{60} \\\text{*If } \alpha \text{ and } \beta \text{ are the roots of the equation } ax^2 + bx + c = 0; a\neq 0 \text{ then} \\\text{ i) } \alpha + \beta = -\frac{b}{a} =- \frac{\text{ coefficient of x}}{\text{coefficient of }x^2} \\\text{ ii) } \alpha . \beta =\frac{c}{a}= \frac{\text{ coefficient term}}{\text{coefficient of }x^2}\\\text{The quadratic equation whose roots are } \alpha \text{ and } \beta \text{ is }(x-\alpha)(x-\alpha)=0 \\\text{ i.e }x^2-(\alpha + \beta)x+\alpha .\beta=0 \\\text{ i.e }x^2-Sx+P=0 \\\text{, where S = Sum of the roots and P = Product of the roots}$

The roots of the quadratic equation

$\dpi{80} \\\text{* If } \alpha \text{ and } \beta \text{ are the roots of the equation } ax^2 + bx + c = 0; a\neq 0 \text{ then} \\\text{ i) } \alpha + \beta = -\frac{b}{a} =- \frac{\text{ coefficient of x}}{\text{coefficient of }x^2}\\\ \\\text{ ii) } \alpha . \beta =\frac{c}{a}= \frac{\text{ coefficient term}}{\text{coefficient of }x^2}\\$

$\dpi{80} \\\text{*If } \alpha \text{ and } \beta \text{ are the roots of the equation } ax^2 + bx + c = 0; a\neq 0 \text{ then} \\\text{ i) } \alpha + \beta = -\frac{b}{a} =- \frac{\text{ coefficient of x}}{\text{coefficient of }x^2} \\\text{ ii) } \alpha . \beta =\frac{c}{a}= \frac{\text{ coefficient term}}{\text{coefficient of }x^2}\\\text{The quadratic equation whose roots are } \alpha \text{ and } \beta \text{ is }(x-\alpha)(x-\alpha)=0 \\\text{ i.e }x^2-(\alpha + \beta)x+\alpha .\beta=0 \\\text{ i.e }x^2-Sx+P=0 \\\text{, where S = Sum of the roots and P = Product of the roots}$

Complex Number :The roots of the quadratic equation

$\dpi{100} \dpi{80} \\\text{1) If a+ib = 0 where i =}\sqrt{-1}\text{ , then a = b = 0} \\\text{2) If a+ib = x+iy where i =}\sqrt{-1}\text{ , then a =x and b = y}\\\text{3) If a+ib = 0 where i =}\sqrt{-1}\text{ , then a = b = 0} \\\text{4) If a+ib = x+iy where i =}\sqrt{-1}\text{ , then a =x and b = y} \\\text{{\color{Red} The roots of the quadratic equation}} \\\text{1) } ax^2+bx+c = 0 ;\;\; a\neq 0 \text{ are}\frac{-b\pm \sqrt{b^2 - 4ac}}{2a} \\\text{where } \Delta \text{ = discriminant = } {\color{Red} b^2 -4ac} \\\ \\\text{ Case - I The roots are real and distinct if }\Delta>0 \\\text{ Case - II The roots are real and coincident if }\Delta = 0 \\\text{ Case - III The roots are non-real if }\Delta < 0$

Logarithm :

$\dpi{80} \\\text{If a,m, n are positive real numbers and a}\neq 1\text{ then }\\\ log_a{mn}=log_a{m}+log_a{n}$

$\dpi{80} \\\text{If a,m, n are positive real numbers and a}\neq 1\text{ then}\\\ log_a{m^n} = nlog_a{m}$

$\dpi{100} \dpi{100} \dpi{80} \\\text{If a,m, n are positive real numbers and a}\neq 1\text{ then }\\\ log_a{mn}=log_a{m}+log_a{n}$

$\dpi{80} \\\text{If a,m, n are positive real numbers and a}\neq 1\text{ then}\\\ log_a{m^n} = nlog_a{m}$

$\dpi{80} \\\text{If a , b and k are positive real numbers}b\neq 1 ,k\neq 1 \text{ then,} \\\log_b{a}=\frac{log_k{a}}{log_k{b}}$

$\dpi{80} log_b{a} = \frac{1}{log_a{b}}\text{ , where a and b are positive real numbers , }a\neq 1 , b\neq 1$

$\dpi{80} \\\text{if a , m , n are positive real numbers },a\neq 1 \text{ and if }log_a{m} = log_a{n} \\\text{Then m = n}$

Algebra :

$\dpi{80}\\\text{1. }(a^m)^n = a^{mn} = (a^n)^m \\\text{2. }(ab)^n = a^n.b^n \\\text{3. }\left ( \frac{a}{b} \right )^n = \frac{a^n}{b^n}\\\text{4. }(a^m)^n = a^{mn} = (a^n)^m \\\text{5. }(ab)^n = a^n.b^n \\\text{6. }\left ( \frac{a}{b} \right )^n = \frac{a^n}{b^n} \\\text{7. } a^0 = 1 \text{ , where a}\in R ,a\neq 0 \\\text{8. }a^{-n} = \frac{1}{a^{n}} \\\text{9. }a^{\frac{p}{q}}=\sqrt[q]{a^{p}}$

Algebra :

$\dpi{100} \dpi{80} \\\text{1. }(a+b)^2 = a^2+2ab+b^2 \\\text{2. }a^2+b^2 = (a+b)^2-2ab\\\text{3. }(a-b)^2 = a^2-2ab+b^2\\\text{4. }a^2+b^2 = (a-b)^2+2ab \\\text{5. }(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)$

Algebra :

$\dpi{80} \\\text{1. }a^n- b^n = (a- b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + ··· + b^{n-1})\\\text{2. }a^n = \text{a.a.a . . . n times} \\\text{3. }a^m.a^n = a^{m+n} \\\text{4. }\frac{a^m}{a^n} = a^{m-n} \text{if m}>n\; \\= 1 \text{ If m = n} \\= \frac{1}{a^{m-n}} \text{ If m }< n;\text{ a}\in \text{ R ;a}\neq 0$

Algebra :

$\dpi{70} \\\text{1. }a^3 +b^3 = (a+b)^3-3ab(a+b)$

$\dpi{70} \\\text{2. }(a-b)^3 = a^3-b^3-3ab(a-b) \\\text{3. }a^3-b^3 = (a-b)^3+3ab(a-b) \\\text{4. }a^2-b^2 = (a+b)(a-b) \\\text{5. }a^3 - b^3 = (a-b)(a^2 + ab + b^2) \\\text{6. }a^3 + b^3 = (a + b)(a^2- ab + b^2)$

For a geometric progression (G.P.) :

$\\\text{i) }n^{th}\text{ term }t_n = a\gamma ^{n-1}$

$\\\text{i) }n^{th}\text{ term }t_n = a\gamma ^{n-1}$

$\\\text{ii) The sum of the first (n) terms :}$

$\\S_n = \frac{a(1-\gamma ^n)}{1-\gamma },\text{ If }\gamma <1 \\ =\frac{a(\gamma ^n-1)}{\gamma -1},\text{ If }\gamma >1 \\=na \text{, If }\gamma =1$

Sum of the sequence of series :

$\dpi{70} \dpi{70} \dpi{70} \dpi{70} \dpi{70} \\\text{i) }\sum_{\gamma =1}^{n}\gamma =1+2+3+...+n = \frac{n}{2}\left ( n+1 \right ) \\\text{ii) }\sum_{\gamma =1}^{n}\gamma ^2=1^2+2^2+3^2+...+n^2 = \frac{n}{6}\left ( n+1 \right )\left ( 2n+1 \right )$

$\dpi{70} \dpi{70} \dpi{70} \dpi{70} \\\text{i) }\sum_{\gamma =1}^{n}\gamma =1+2+3+...+n = \frac{n}{2}\left ( n+1 \right ) \\\text{ii) }\sum_{\gamma =1}^{n}\gamma ^2=1^2+2^2+3^2+...+n^2 = \frac{n}{6}\left ( n+1 \right )\left ( 2n+1 \right ) \\\text{iii) }n! = (1).(2).(3)....(n-1).n \\\text{iv) }n! =n(n-1)! = n(n-1)(n-2)!=...... \\\text{v) }0! =1 \\\text{vi) }\left ( a+b \right )^n=a^n=na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+\frac{n(n-1)(n-2)}{3!}a^{n-3}b^3+....+b^n,n>1$

VBODMAS Rules

$\\\text{V }\Rightarrow \text{Vinculum or Bar , You can see on top of the values like }\bar{X} \\\text{B }\Rightarrow \text{Bracket like() }\left \{ \right \}\left [ \right ] \\\text{O }\Rightarrow \text{Means of} \\\text{D }\Rightarrow \text{Means Division} \\\text{M }\Rightarrow \text{Means Multiplication} \\\text{A}\Rightarrow \text{Means Addition} \\\text{S }\Rightarrow \text{Means Subtraction}$

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