Algebra Review

ALGEBRA HINTS
Order of Operations (PEMDAS)
PLEASE EXCUSE MY DEAR AUNT SALLY
Parentheses Exponents Mult. Div. Add. Subt.
    Left to right, perform either multiplication OR division Left to right, perform either addition OR subtraction

 

Example:   $ \displaystyle 20-16\div {{2}^{3}}\times \left( {3+2} \right)=20-\frac{{16}}{8}\times 5=20-\left( {2\times 5} \right)=10$

 

Properties of Real Numbers

Associative (groups): $ a+\left( {b+c} \right)=\left( {a+b} \right)+c,\,\,\,\,a\left( {bc} \right)=\left( {ab} \right)c$

Commutative (ordering): $ a+b=b+a,\,\,\,\,ab=ba$

Distributive (“pushing through”): $ a\left( {b+c} \right)=ab+ac,\,\,\,\,\,a\left( {b-c} \right)=ab-ac$

Additive Inverse: $ \displaystyle a+-a=0$    Multiplicative Inverse: $ \displaystyle a\times \frac{1}{a}=1$

Additive Identity: $ a+0=a$     Multiplicative Identity: $ a\times 1=a$

Zero Property: $ a\times 0=0$

Scientific Notation

Examples: $ \displaystyle \begin{array}{c}7.9\times {{10}^{4}}=79,000\\7.9\times {{10}^{{-4}}}=.00079\end{array}$

Count to the right (left) that many decimal places when the exponent is positive (negative). Note that the first number must be $ \ge 1$ and $ <10$.

Operations

$ \begin{align}a\left( {x+y} \right)=ax+ay\\\frac{{a+b}}{c}=\frac{a}{c}+\frac{b}{c}\\\frac{a}{b}+\frac{c}{d}=\frac{{ad+bc}}{{bd}}\\\frac{a}{b}\times \frac{c}{d}=\frac{{ac}}{{bd}}\\\frac{{\frac{a}{c}}}{{\frac{b}{d}}}=\frac{a}{c}\div \frac{b}{d}=\frac{a}{c}\times \frac{d}{b}=\frac{{ad}}{{cb}}\end{align}$

Factorial: $ 6!=6\times 5\times 4\times 3\times 2\times 1=720;\,\,\,\,\,1!=1;\,\,\,\,\,\,0!=1$

Absolute Value: $ \left| 5 \right|=5;\,\,\,\,\,\,\left| {-5} \right|=5$

Exponents and Radicals

$ \displaystyle \begin{align}{{(xy)}^{m}}&={{x}^{m}}\cdot {{y}^{m}}\\{{\left( {\frac{x}{y}} \right)}^{m}}&=\frac{{{{x}^{m}}}}{{{{y}^{m}}}}\\{{x}^{m}}\cdot {{x}^{n}}&={{x}^{{m+n}}}\\\frac{{{{x}^{m}}}}{{{{x}^{n}}}}&={{x}^{{m-n}}}\\{{\left( {{{x}^{m}}} \right)}^{n}}&={{x}^{{mn}}}\\{{x}^{1}}&=x;\,\,\,\,{{x}^{0}}=1\\\frac{1}{{{{x}^{m}}}}&={{x}^{{-m}}}\\{{\left( {\frac{x}{y}} \right)}^{{-m}}}&={{\left( {\frac{y}{x}} \right)}^{m}}\end{align}$      $ \displaystyle \begin{align}\sqrt[n]{x}&={{x}^{{\frac{1}{n}}}}\,\,\\\sqrt[n]{{xy}}&=\sqrt[n]{x}\cdot \sqrt[n]{y}\,\,\left( {\text{Real numbers only}} \right)\\{{\left( {\sqrt[n]{x}} \right)}^{m}}&=\,\sqrt[n]{{{{x}^{m}}}}={{x}^{{\frac{m}{n}}}}\\\sqrt[n]{{{{x}^{n}}}}&=\,\left| x \right|\\\frac{x}{{\sqrt{y}}}&=\frac{x}{{\sqrt{y}}}\cdot \frac{{\sqrt{y}}}{{\sqrt{y}}}=\frac{{x\sqrt{y}}}{y}\,\,(\text{Rationalize})\\&\text{Use Conjugate}:\\\frac{x}{{x+\sqrt{y}}}&=\frac{x}{{x+\sqrt{y}}}\cdot \frac{{x-\sqrt{y}}}{{x-\sqrt{y}}}=\frac{{x\left( {x-\sqrt{y}} \right)}}{{{{x}^{2}}-y}}\end{align}$

Lines

Slope of line going through points $ \left( {{{x}_{1}},{{y}_{1}}} \right)$ and $ \left( {{{x}_{2}},{{y}_{2}}} \right)$ is $ \displaystyle m=\frac{{{{y}_{2}}-{{y}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}}$. Point-slope equation of line through point $ \left( {{{x}_{1}},{{y}_{1}}} \right)$ with slope $ m$ is $ y-{{y}_{1}}=m\left( {x-{{x}_{1}}} \right)$. Slope-intercept equation of line with slope $ m$ and $ y$-intercept $ b$ is $ y=mx+b$.

 

Examples: Find the equation that passes through points $ \left( {4,0} \right)$ and $ \left( {-2,12} \right)$. First find slope: $ \displaystyle m=\frac{{{{y}_{2}}-{{y}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}}=\frac{{12-0}}{{-2-4}}=-2$.

For point-slope equation, use either point: $ y-0=-2\left( {x-4} \right);\,\,\,y=-2\left( {x-4} \right)=-2x+8$.

For slope-intercept equation, find $ y$-intercept ($ m$), using either point: $ y=-2x+b;\,\,12=-2\left( {-2} \right)+b;\,\,\,b=8:\,\,\,y=-2x+8$.

 

Example of slope of 0: line $ y=5$ (horizontal line). Example of undefined slope: $ x=-3$ (vertical line).

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other, for example $ 4$ and $ \displaystyle -\frac{1}{4}$.

Standard Form of Line is $ Ax+By=C$, where $ A$ is a positive integer. For example, $ \displaystyle y=-\frac{1}{3}x+4$ in standard form is $ x+3y=12$.

Solving Linear Equations

1. Combine like terms.

2. Get all variables on left, constants on right.

3. Isolate the variable by dividing what’s in front of it.

Example: $ \displaystyle \begin{align}5x-2x-5&=16\\3x-5&=16\\3x-5+5&=16+5\end{align}$      $ \begin{align}3x&=21\\\frac{{3x}}{3}&=\frac{{21}}{3}\\x&=7\end{align}$

 

For Inequalities, remember to change the sign when multiplying or dividing by a negative number!

For Systems of Equations, use either substitution (substitute the expression of one variable into the other equation) or linear elimination to first solve for one variable, then the other. Example of elimination:

$ \displaystyle \begin{array}{l}2x+5y=-1\,\,\,\,\,\text{multiply by}-3\\7x+3y=11\text{ }\,\,\,\,\,\text{multiply by }5\\-6x-15y=3\,\\\,\underline{{35x+15y=55}}\text{ }\\\,29x\,\,\,\,\,\,\,\,\,\,\,\,\,\,=58\\\,\,\,\,\,\,\,\,\,\,\,x=2\end{array}$

Use either equation to get $ y$: $ \displaystyle y=\frac{{-1-2\left( 2 \right)}}{5}=-1$. Solution is $ \left( {2,-1} \right)$.

Direct/Indirect Variation

Direct Variation formula: $ \displaystyle y=kx$ or $ \displaystyle  \frac{{{{y}_{1}}}}{{{{x}_{1}}}}=\frac{{{{y}_{2}}}}{{{{x}_{2}}}}$        ($ k$ is a constant)

Example: $ y$ varies directly with $ x$. $ y=10$ when $ x=2$. What is $ y$ when $ x=4$?

$ y=kx;\,\,10=k\left( 2 \right);\,\,k=5.\,\,\,\,\,y=5x;\,\,\,y=5\left( 4 \right)=20$

(Or, $ \displaystyle \frac{{10}}{2}=\frac{{{{y}_{2}}}}{4};\,\,2{{y}_{2}}=40;\,\,{{y}_{2}}=20$)

Indirect or Inverse Variation formula: $ \displaystyle y=\frac{k}{x}$, or $ k=xy$        ($ k$ is a constant)

Example: $ y$ varies indirectly with $ x$. $ y=10$ when $ x=4$. What is $ y$ when $ x=2$?

$ \displaystyle y=\frac{k}{x};\,\,10=\frac{k}{4};\,\,k=40.\,\,\,\,\,y=\frac{{40}}{x};\,\,\,y=\frac{{40}}{2}=20$

Joint Variation: Like direct variation, but involves more than one variable. Example: $ y=kx{{z}^{2}}$

Combined Variation: Combination of direct or joint variation, and indirect variation. Example: $ \displaystyle y=\frac{{kxw}}{{{{z}^{2}}}}$

Partial Variation: Variables are related by sum of two or more variables. Example: $ \displaystyle y={{k}_{1}}x+\frac{{{{k}_{2}}}}{x}$

Multiplying/Factoring Polynomials

Multiplying Binomials:

$ \begin{array}{c}\left( {a+b} \right)\left( {c+d} \right)=ac+ad+bc+bd\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F\,\,\,\,\,\,\,O\,\,\,\,\,\,\,I\,\,\,\,\,\,\,L\end{array}$

Perfect Square Binomials:

$ \begin{array}{c}{{\left( {a+b} \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\\{{\left( {a-b} \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\end{array}$

Difference of Squares:

$ \displaystyle \left( {a+b} \right)\left( {a-b} \right)={{a}^{2}}-{{b}^{2}}$

Sum/Difference of Cubes:

$ \displaystyle \begin{array}{c}{{\left( {a+b} \right)}^{3}}=\left( {a+b} \right)\left( {{{a}^{2}}-ab+{{b}^{2}}} \right)\\{{\left( {a-b} \right)}^{3}}=\left( {a-b} \right)\left( {{{a}^{2}}+ab+{{b}^{2}}} \right)\end{array}$

Quadratics
Quadratic Formula:

Quadratic Formula gives roots of $ a{{x}^{2}}+bx+c=0$:

$ \displaystyle \frac{{-b\pm \sqrt{{{{b}^{2}}-4ac}}}}{{2a}}$

Example:  For $ {{x}^{2}}-4x-12$, $ a=1,\,\,b=-4,\,\,c=-12$.

Roots are:

$ \displaystyle \begin{align}x&=\frac{{-\left( {-4} \right)\pm \sqrt{{{{{\left( {-4} \right)}}^{2}}-4\left( 1 \right)\left( {-12} \right)}}}}{{2\left( 1 \right)}}=\frac{{4\pm \sqrt{{64}}}}{2}=\frac{{4\pm 8}}{2}\\x&=6,\,-2\end{align}$

 

To get vertex, use $ \displaystyle \left( {-\frac{b}{{2a}},\,f\left( {-\frac{b}{{2a}}} \right)} \right)$ (plug in what you get for $ x$ to get $ y$). $ \displaystyle x=-\frac{b}{{2a}}$ is the axis of symmetry. In above example, axis of symmetry is $ \displaystyle x=-\frac{b}{{2a}}=-\frac{{-4}}{{2\left( 1 \right)}}=2;\,\,\,\,x=2$. The vertex is $ \displaystyle \left( {2,{{{\left( 2 \right)}}^{2}}-4\left( 2 \right)-12} \right)=\left( {2,-16} \right)$.

 

Vertex Form: $ \displaystyle y=a{{(x-h)}^{2}}+k,\,\,\text{or }y-k=a{{(x-h)}^{2}}$, vertex is $ \left( {h,\,k} \right)$.

 

To Complete the Square to go from Standard Form to Vertex Form, halve the coefficient of $ x$, square it, add it to complete the square, and then subtract it; for example:

$ \begin{align}y&={{x}^{2}}-4x-12\\y&=\left( {{{x}^{2}}-4x+\,\,\color{green}{{\underline{{{{{\left( {\frac{4}{2}} \right)}}^{2}}}}}}\,} \right)-12-\color{green}{{\underline{{{{{\left( {\frac{4}{2}} \right)}}^{2}}}}}}\\y&=\left( {{{x}^{2}}-4x+\,\,4\,} \right)-12-4\\y&={{\left( {x-2} \right)}^{2}}-16\,\,\,\,\,\,\,\text{Vertex:}\,\,\,\left( {2,-16} \right)\end{align}$

 

Factored Form: $ a\left( {x-{{r}_{1}}} \right)\left( {x-{{r}_{2}}} \right)=0$. Roots are $ ({{r}_{1}},0),\,\,\,({{r}_{2}},0)$.

Function Characteristics

Note that for a relation (equation) to be a function, it must pass the vertical line test, meaning that you can’t have more than one $ y$-value for any $ x$-value. For example, vertical lines aren’t functions, and neither are the points $ \left( {2,-5} \right)\,\,\text{and}\,\,\left( {2,0} \right)$.

Domain Restrictions (Undefined):

Domains are restricted to the following:

  • Denominator can’t be $ 0$: $ \displaystyle \frac{a}{{\ne 0}}$
  • Even root must be $ \ge 0$:  $ \sqrt[{\text{even}\,}]{{\,\ge 0}}$
  • Argument of log must be $ >0$: $ \log \left( {>0} \right)$ or $ \ln \left( {>0} \right)$

Examples: The domain of $ f\left( x \right)=\sqrt{{x+3}}$ is $ x+3\ge 0$, or $ x\ge -3$. The domain of $ f\left( x \right)=\log \left( {3x-2} \right)$ is $ 3x-2>0$or $ x>\frac{2}{3}$The domain of $ f\left( x \right)=\frac{3}{{\sqrt{x}}}$ is $ x\ge 0\,\,\,\text{and}\,\,\,x\ne 0$, or $ x>0$.

Note that for a relation (equation) to be a function, it must pass the vertical line test, meaning that you can’t have more than one $ y$-value for any $ x$-value. For example, vertical lines aren’t functions, and neither are the points $ \left( {2,-5} \right)\,\,\text{and}\,\,\left( {2,0} \right)$.

Inverse Functions

To get inverse function, switch the $ x$ and $ y$ variables and solve for the “new” $ y$. Note also that if two functions $ f$ and $ g$ are inverses, $ f\left( {g\left( x \right)} \right)=g\left( {f\left( x \right)} \right)=x$. (These are composites of functions: put inside function everywhere there’s an “$ x$” in the outside function.)

Example: $ x=3{{y}^{2}}+5;\,\,{{y}^{2}}=\frac{{x-5}}{3};\,\,{{f}^{{-1}}}\left( x \right)=\pm \sqrt{{\frac{{x-5}}{3}}}$. (This inverse is not a function; to restrict to function, it’d be positive only.)  Also, $ f\left( {{{f}^{{-1}}}\left( x \right)} \right)=3{{\left( {\sqrt{{\frac{{x-5}}{3}}}} \right)}^{2}}+5=x$ and $ {{f}^{{-1}}}\left( {f\left( x \right)} \right)=\sqrt{{\frac{{\left( {3{{x}^{2}}+5} \right)-5}}{3}}}=x$.

Transformations of Functions

To shift functions, make vertical shifts up and down based on what is added/subtracted on the outside, and vertical stretches/compressions based on what is multiplied/divided on the outside. Make horizontal shifts in opposite direction based on what is added/subtracted on the inside, and horizontal stretches/compressions in oppositive direction based on what is multiplied/divided on the inside. Perform stretches/compressions before shifts.

Example: For $ y=\color{green}{2}{{\left( {\color{blue}{3}\left( {x\color{blue}{{-4}}} \right)} \right)}^{2}}\color{green}{{+5}}$ (original function $ y={{x}^{2}}$), $ y$-coordinates become $ 2y+5$, and $ x$-coordinates become $ \frac{1}{3}x+4$.

Degrees of Polynomials

To get the degree of a polynomial, add up all the exponents of each term (when multiplied out), and the degree is the highest number.

 

For example, for the polynomial $ 2{{x}^{3}}{{y}^{2}}+x{{y}^{2}}-xy+6$, the degree is $ 5$, since the first term’s sum of exponents is $ 5$, the second is $ 3$, the third is $ 2$, and the last is $ 0$.

 

For the polynomial $ x\left( {2x-3} \right)\left( {{{x}^{2}}+1} \right)$, the degree is $ 4$, since the term with the highest exponent would be $ x\cdot 2x\cdot {{x}^{2}}=2{{x}^{4}}$.

Factoring Quadratics and Roots

“FOIL” (multiply binomials) backwards! Example: To factor $ {{x}^{2}}-4x-12$, find two numbers that multiply to $ -12$ but add to $ -4$: $ -6$ and $ 2$. Check, and it works: $ \begin{array}{l}\,\,\,\,\,{{x}^{2}}-4x-12\\=\left( {x-6} \right)\left( {x+2} \right)\end{array}$. To get roots, set each factor to $ 0$ and solve: $ \begin{array}{l}x-6=0;\,\,x=6\\x+2=0;\,\,x=-2\end{array}$.

 

To factor a quadratic like $ 3{{x}^{2}}+10x-8$, either use “guess and check” or “ac” method. For the “ac” method, find two numbers that multiply to $ -24$ ($ 3$ times  $ -8$), but add to $ 10$ (coefficient of middle term): $ 12$ and  $ -2$. Separate and factor:

$ \begin{array}{c}3{{x}^{2}}+10x-8\\3{{x}^{2}}\,\,\,\,\,+12x-2x\,\,\,\,\,\,-8\\3x\left( {x+4} \right)-2\left( {x+4} \right)\\\left( {3x-2} \right)\left( {x+4} \right)\end{array}$

Permutations/Combinations

Permutations:

Number of $ n$ things taken $ r$ at a time, order matters:

$ \displaystyle {}_{n}{{P}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!}}$

(On TI Graphing Calculator, for example, for $ {}_{{10}}{{P}_{3}}$, press $ 10$, then MATH, then Right Arrow three times to PROB, then $ 2$, for nPr, then $ 3$.)

(Example: the number of ways to select a President and Vice-President out of a group of 10 people is $ {}_{{10}}{{P}_{2}}$.)

 

Combinations:

Number of $ n$ things taken $ r$ at a time, order doesn’t matter:

$ \displaystyle {}_{n}{{C}_{r}}=\frac{{n!}}{{r!\left( {n-r} \right)!}}$

(Example: the number of ways to select two officers out of a group of 10 people is $ {}_{{10}}{{C}_{2}}$.)

Exponential Growth/Decay

Exponential Growth: $ y=a{{b}^{x}},\,\,a>0,\,\,b>1$

Exponential Decay: $ y=a{{b}^{x}},\,\,a>0,\,\,0<b<1$

 

Compounding Formulas: $ A$ ending amount, $ P$ beginning amount, $ r$ annual rate, $ n$ number of times compounded per year, $ t$ number of years:

Growth: $ \displaystyle A=P{{\left( {1+\frac{r}{n}} \right)}^{{nt}}}$     Decay: $ \displaystyle A=P{{\left( {1-\frac{r}{n}} \right)}^{{nt}}}$

 

Simple Interest (linear, not exponential) growth: $ \displaystyle A=P\left( {1+rt} \right)$, Interest only is $ I=Prt$.

 

Continuous compounding ($ e$ is Euler’s number ): $ A=P{{e}^{{rt}}}$

 

Half Life: $ y=a{{\left( {.5} \right)}^{{\frac{t}{p}}}}$, $ a$ beginning amount, $ y$ ending amount, $ t$ time, $ p$ half-life (time it takes to halve)

Logarithms

$ y={{\log }_{b}}x\,\,\,\,\text{means}\,\,\,\,x={{b}^{y}}\,\,\,(b>0,\,\,b\ne 1,\,\,x>0)$

$ \displaystyle \begin{array}{c}{{\log }_{a}}a=1\\{{\log }_{a}}{{a}^{x}}=x\\{{\log }_{a}}1=0\\{{a}^{{{{{\log }}_{a}}x}}}=x\end{array}$                 $ \begin{array}{c}\ln e\,\,\left( {={{{\log }}_{e}}\left( e \right)} \right)=1\\\ln {{e}^{x}}\,\,=x\\\ln \left( 1 \right)\,\,=0\\{{e}^{{\ln x}}}=x\end{array}$

 

Log Rules:

$ \begin{align}{{\log }_{b}}\left( {xy} \right)&={{\log }_{b}}x+{{\log }_{b}}y\\{{\log }_{b}}\left( {\frac{x}{y}} \right)&={{\log }_{b}}x-{{\log }_{b}}y\\{{\log }_{b}}\left( {{{x}^{p}}} \right)&=p{{\log }_{b}}x\end{align}$             $ \displaystyle \begin{align}{{\log }_{b}}\left( x \right)&=\frac{{{{{\log }}_{{10}}}x}}{{{{{\log }}_{{10}}}b}}\\{{\log }_{b}}\left( x \right)&=\frac{{\ln x}}{{\ln b}}\end{align}$

 

Take the ln or log of each side and then use the Power Rule when solving for variables in exponents. For example,

$ \displaystyle \begin{align}150{{\left( {.5} \right)}^{t}}&=1500\\{{\left( {.5} \right)}^{t}}&=\frac{{1500}}{{150}}=10\end{align}$          $ \displaystyle \begin{align}\ln {{\left( {.5} \right)}^{t}}&=\ln \left( {10} \right)\\t\ln \left( {.5} \right)&=\ln \left( {10} \right)\\t&=\frac{{\ln \left( {10} \right)}}{{\ln \left( {.5} \right)}}\approx -3.32\end{align}$

Distance/Midpoint Formulas

Midpoint between points $ \left( {{{x}_{1}},{{y}_{1}}} \right)$ and $ \left( {{{x}_{2}},{{y}_{2}}} \right)$ is:

$ \displaystyle \left( {\frac{{{{x}_{1}}+{{x}_{2}}}}{2},\frac{{{{y}_{1}}+{{y}_{2}}}}{2}} \right)$

Distance between points $ \left( {{{x}_{1}},{{y}_{1}}} \right)$ and $ \left( {{{x}_{2}},{{y}_{2}}} \right)$ is:

$ d=\sqrt{{{{{\left( {{{x}_{2}}-{{x}_{1}}} \right)}}^{2}}+{{{\left( {{{y}_{2}}-{{y}_{1}}} \right)}}^{2}}}}$

 

Other formulas:

Pythagorean Theorem:   $ {{a}^{2}}+{{b}^{2}}={{c}^{2}}$

Equation of Circle with center $ (h,k)$ and radius $ r$:  $ {{\left( {x-h} \right)}^{2}}+{{\left( {y-k} \right)}^{2}}={{r}^{2}}$

Formulas for Area, Circumference, Surface Area and Volume:

Triangle: $ \displaystyle A=\frac{1}{2}bh$    Circle:  $ A=\pi {{r}^{2}};\,\,C=2\pi r$

Sphere: $ \displaystyle SA=4\pi {{r}^{2}};\,\,\,V=\frac{4}{3}\pi {{r}^{2}}$

Cylinder: $ \displaystyle SA=2\pi {{r}^{2}}+2\pi rh;\,\,\,V=\pi {{r}^{2}}h$

Cone:  $ \displaystyle V=\frac{1}{3}\pi {{r}^{2}}h$

Complex/Imaginary Numbers

$ {{i}^{2}}=-1;\,\,\,\,\,\,i=\sqrt{{-1}}$

 

Example operations:

$ \displaystyle 6+i-\left( {4-2i} \right)=2+3i$

$ \displaystyle \require {cancel} \begin{align}\left( {3+2i} \right)\left( {3-2i} \right)&=9\cancel{{-6i}}\cancel{{+6i}}-{{\left( {2i} \right)}^{2}}\\&=9-4{{i}^{2}}=9-4\left( {-1} \right)=13\end{align}$

 

Convert to $ i$ first:

$ \begin{align}\sqrt{{-3}}\cdot \sqrt{{-24}}&=\sqrt{3}i\cdot \sqrt{{24}}i=\sqrt{3}i\cdot \sqrt{4}\cdot \sqrt{6}i\\&=\sqrt{3}i\cdot 2\sqrt{6}i=2\sqrt{{3\cdot 6}}\cdot {{i}^{2}}=2\sqrt{{18}}\cdot {{i}^{2}}\\&=-2\cdot \sqrt{9}\cdot \sqrt{2}=-2\cdot 3\sqrt{2}=-6\sqrt{2}\end{align}$

 

To rationalize by using conjugate:

$ \displaystyle \frac{4}{{1+i}}=\,\frac{4}{{1+i}}\cdot \frac{{1-i}}{{1-i}}=\frac{{4-4i}}{{1-{{i}^{2}}}}=\frac{{4-4i}}{{1+1}}=2-2i$

Sequences and Series:

$ n=$ term number, $ {{a}_{n}}=nth$ term, $ {{a}_{1}}=$ first term, $ d=$ common difference (2nd term minus 1st in arithmetic sequence), $ r=$ common ratio (2nd term divided by 1st in geometric sequence)

$ {{S}_{n}}=$ sum of first $ n$ terms, $ {{S}_{\infty }}=$ sum of infinite number of terms.

Arithmetic:

$ {{a}_{n}}={{a}_{1}}+\left( {n-1} \right)d$

(Recursive: $ {{a}_{n}}=a,\,\,{{a}_{n}}={{a}_{{n-1}}}+d$)

$ \displaystyle {{S}_{n}}=\frac{n}{2}\left( {{{a}_{1}}+{{a}_{n}}} \right)\,\,\,=\,\,\,\frac{n}{2}\left( {2{{a}_{1}}+\left( {n-1} \right)d} \right)$

Geometric:

$ {{a}_{n}}={{a}_{1}}{{\left( r \right)}^{{n-1}}}$(Recursive: $ {{a}_{n}}=a,\,\,{{a}_{n}}=r{{a}_{{n-1}}}$)

$ \displaystyle {{S}_{n}}=\frac{{{{a}_{1}}\left( {1-{{r}^{n}}} \right)}}{{1-r}}\,\,\,\,\,\,{{S}_{\infty }}=\frac{{{{a}_{1}}}}{{1-r}},\,\,\,\,\,\,\,\left| r \right|<1$

 

Example: The sum of the first $ 5$ terms of the geometric $ 1,2,4,8,…$ is?

$ \displaystyle r=\frac{{\text{second}}}{{\text{first}}}=\frac{{\text{third}}}{{\text{second}}}=2;\,\,{{a}_{1}}=1;\,\,n=5;\,\,{{S}_{5}}=\frac{{1\left( {1-{{2}^{5}}} \right)}}{{1-2}}=31$

$ (1+2+4+8+16=31\,\,\,\,\surd )$