Math (GRE)

Formula for arithmetic sequence (7!)

an=a1+(n-1)d

Formula for geometric sequence (68!)

an = a1 * r^(n-1)

formula for Sum of the first n terms of a geometric sequence (32!)

Sn=a1(1-r^n)/(1-r)

formula for Sum of the first n terms of an arithmetic sequence (20!)

Sn=n(a1+an)/2

formula for percent change

difference/original x 100

formula for # of diagonals in n-sided polygon (78!)

n(n-3)/2

formula for volume of a cone (39!)

V=(1/3)πr²h

What is the sum of the exterior angles of a polygon? Why?

360

Because the interior angles sum to 180(n-2) degrees and each exterior angle is, by definition, supplementary to its interior angle.

formula for measure of each exterior angle in an n-sided regular polygon (arguably not very important if you know interior angle?)

360/n (or 180 - 180(n-2)/n)

formula relating LCM and GCF (of 2 numbers)

LCM x GCF (of 2 numbers) = product of the 2 numbers

Formula for the surface area of a sphere

4πr²

formula for volume of a sphere

V=(4/3)πr³ [think volume is in cubic units]

formula for percentiles (5!)

k = n (percentile/100) (either round k up if not integer, or find average of kth & (k+1)th terms if k is integer)

formula for compound interest

Jasmine deposits $520 into a savings account that has a 3.5% interest rate compounded monthly. What will be the balance of Jasmine's savings account after two years? ..................................... A college student expects to earn at least $1,000 in interest on an initial investment of $20,000. If the money is invested for one year at an annual interest rate of r percent, compounded quarterly, what is the least annual interest rate that would achieve the goal? (Give your answer to the nearest 0.1 percent.)

A=P(1+r/n)^(nt)

$557.65 ......................................................................................................4.9%

Formula for vertex of parabola

x=-b/2a

formula for discriminant (and what it means) (17!)

D=b^2-4ac (if negative value, no real roots. if 0, one real, rational root. if positive perfect square, two real, rational roots. if positive non-perfect square, two real, irrational roots)

formula for variance

(SD)^2

Formula for surface area of a cylinder

SA = 2πr(r+h)

formula for area of trapezoid

A=1/2(b1+b2)h

formula for percent profit

[(selling price - cost to buy)/(cost to buy)] x 100

formula for cyclic permutation

r!/r (if 4 elements, r=4, etc) or (r-1)!

formula relating miles to miles per gallon

miles = miles per gallon x gallons

Formula for area of a regular hexagon (or what's the alternative if you forget?)

( 3S^2 sqrt3 ) / 2 (or calculate the total combined area of 6 equilateral triangles if you forget)

formula for geometric mean

x = sqrt(ab)

divisibility rule for 11 (also keep in mind everything from 3 to 12, esp 7) (71!)

a number is divisible by 11 if the positive difference of the two sums of the odd-place and even-place digits of a number is 0 or a multiple of 11.

what to do with repeating portion of decimal (if need to write number as fraction) (28!)

check walmart notebook (keep in mind that repeating number means that there's a factor that isn't 5^x or 2^x such as 1/30 is 1/(5x6))

how are ending zeroes created

(5^x)(2^x)

how to do average speed/miles per gallon problems and parts of resulting mixture problems the correct way

the long way!

Does distribution affect range?

no

regarding data, what does it mean if the middle of the curve is pointier?

there is more data within 1 SD of the mean (than if the curve is more spread out)

4 things you can't assume (regarding geometry) and 2 things you CAN assume (51!)

can't assume: whether polygon is regular, parallel lines, perpendicular lines, right angles ... CAN assume that lines are straight and that a triangle is absolutely a right triangle if it matches up with Pythagorean Theorem

reminder that there can be multiple ways to calculate bxh in a triangle (especially when multiple triangles within a larger triangle)

okay

Does "ABC is a right triangle" tell you which angle is a right angle?

No

Types of Triangle Congruency (and which is Similar and which is none of the above)

SAS, SSS, ASA, AAS, and Hypotenuse-Leg (AAA is similar, ASS is none of the above)

In a population of 500, how many data points are there between 50th and 75th percentile and how many percentiles between the two percentiles?

125, 25

If an integer divides b & c, then what else does it divide?

bx + cy

how to solve "integer has how many positive divisors" problems the infallible way

72 has how many positive integers?

do prime factorization, add 1 to each exponent, and multiply new exponents together

72 = 3 x 3 x 2 x 2 x 2 ... thus 3 x 4 = 12

In 1 revolution, a cylinder rolls the equivalent of what distance?

its circumference

(4^x)(3^x) = ?

12^x

Parallel lines may have what kind of angles equal?

Alternate interior angles (equal)

standard deviation is based on what 2 things

mean and spread (thus, mean must be known to calculate SD)

Are there times when median can't be determined?

Yes

what is the only shape in which there is a correlation between the size of a side and the size of its opposite angle?

Triangle

In a triangle, exterior angle is equal to what sum?

sum of remote interior angles (aka the 2 triangle angles that are far away from the exterior angle)

Central vs Inscribed Angles

when given inscribed angle, double it to convert it to central angle and find area/circumference of sector given angle

inscribed vs circumscribed

inscribed is inside, circumscribed is outside

how can you tell an inscribed triangle is a right triangle?

when it's inscribed in a semicircle or when one of its sides equals the diameter of the circle

how to find LCM of 3 or more numbers

Find LCM of 8, 36, 54

use prime factorization and multiply each factor the maximum number of times it occurs

8 - 2 x 2 x 2, 36 - 3 x 3 x 2 x 2, 54 - 3 x 3 x 3 x 2: LCM - 3 x 3 x 3 x 2 x 2 x 2 = 216

reminder that you can't assume integer (question must specify)

okay

how to convert odd units like gal/mi into mpg

do the reciprocal

The Last digit Shortcut

What is the units digit of (7^2)(9^2)(3^3)?

When asked to find the units digit, just look at the units digit of the product.

For example: What is the units digit of (7^2)(9^2)(3^3)?

Step 1: 7 x 7 = 49 - Drop all except units digit - 9

Step 2: 9 x 9 = 81 - Drop all except units digit - 1

Step 3: 3 x 3 x 3 = 27 - Drop all except units digit - 7

Step 4: 9 x 1 x 7 = 63

The units digit of the final product is 3

Tens Digit Shortcut (and Hundreds, and so on)

What is the tens digit of 12345 x 6789?

If asked for tens, use tens and ones digit (and so on for hundreds, etc)

If asked for the tens digit of 12345 x 6789, do 45 x 89 = 4005 (the tens digit of the final product is 0)

Area relating to aspect ratio

It gets smaller as the aspect ratio gets more extreme from 1 (since square has max area and min perimeter)

When a quadrilateral is inscribed in a circle, opposite angles are _____

supplementary

what number is divisible by any positive or negative integer?

0

reminder to always consider extremes

okay

when does SD change?

with multiplication/division of data set

arrangement vs distinct

arrangement = permutation, but distinct doesn't necessarily mean permutation

if a score is "exactly" a percentile, then...

then K is an integer and someone else scored the same score if all scores are integers

odds of rolling a 3 and a 4 with a pair of dice

2/36 aka (2)(1/6)(1/6)

how to do quad root

how to do eighth root

how to do sixteenth root

...and so on

do square root twice

do square root thrice

do square root four times

...and so on

square root symbol means

positive only

when there are parallel lines, which line with which acute angle connecting them is the longest of lines connecting the parallel lines?

the line with the smallest acute angle (seems counterintuitive)

when equilateral triangle is inscribed in circle, then

then angles are evenly bisected from center of circle

inverse proportions

products are equal (such as more workers takes less time than fewer workers takes)

Extended ratios: if R:W = 2:3 & W:B = 4:5, then R:W:B equals what?

8:12:15

What are the slopes of diagonals of a square

Don't assume plus or minus 1, since it depends on the orientation of the square

Problems like "N is divisible by #1 & #2. Is it divisible by #3?"

N is divisible by 91 and 2. Is it divisible by 26?

N is divisible by 5 and 12. Is it divisible by 24?

Yes if #3's primes are all cancelled out by #1's/#2's, otherwise you can't tell

Yes, since 26's 2 and 13 are cancelled out by 91's 13 and 2

No, since while 2 of 24's 2 and its 3 are cancelled out by 12's 2 2s and 3, 24's third 2 isn't able to be cancelled out.

feel free to convert inequalities (?)

okay

pay attention to the words "different" and/or "unique"

okay

What is 10% of 20% of 30%?

0.3 x 0.2 x 0.1 = Ans

if Answers are in percents, then multiply by 100

For equilateral triangle ABC, what's the probability that the altitude drawn from A to line BC is congruent to the line segment drawn from A to midpoint of line BC? What about for non-equilateral isosceles triangle ABC?

association between equilateral and isosceles triangles

1, 1/3,

equilateral triangles are also isosceles

If a man leaves camp, then travels 4 mi to river, then rides 5 miles, how far (x) is he from camp?

| x - 5 | ≤ x ≤ | x + 5 |

formula for how the perimeters and areas of 2 similar shapes relate?

the ratio of 2 similar shapes' areas = (the ratio of their perimeters) squared

how does a radius that bisects a chord relate to the chord

perpendicular

how to calculate (1,000,001) squared - (999,999) squared

think a squared minus b squared

what does 5^(k+1) equal

(5^k)5

how to maximize area

think perpendicular (or square)

5 properties that squares and rhombuses share vs major difference between squares and rhombuses (other than the obvious) (13!)

1. all sides congruent

2. diagonals perpendicular

3. diagonals bisect angles

4. A = 0.5 (product of diagonals)

5. diagonals bisect each other

…compare these with rectangles

vs Diagonals are only congruent in a square, not in a rhombus

2 properties that rectangles have (and 2 they don’t!) (86!)

1. diagonals are congruent

x. diagonals NOT perpendicular

x. diagonals NOT bisect angles

2. diagonals bisect each other

5 properties that all parallelograms have (and 1 only some have) (79!)

1. 2 pairs of parallel lines and 2 pairs of congruent lines (can also be 1 pair of congruent AND parallel lines)

2. consecutive angles are supplementary (if 1 pair is supplementary, so is the other) and opposite angles are equal

3. diagonals bisect each other

4. diagonals divide parallelogram into 2 congruent triangles (thus, alternate interior angles are equal when making diagonals

5. sum of squares of sides = sum of squares of diagonals

(diagonals only bisect angles if it’s a square or rhombus)

5 properties of a kite (73!)

1. The diagonals are perpendicular.

2. The longer diagonal of a kite bisects the shorter one.

3. 2 pairs of congruent sides

4. Exactly 1 pair of congruent angles (LRnon-vertex angles)

5. Non-Congruent (vertex) angles bisected by long diagonal

2 properties of all trapezoids (82!)

1. 1 pair of parallel lines

2. leg angles are supplementary

4 properties of isosceles trapezoid (80!)

1. legs are congruent

2. base angles are congruent

3. diagonals are congruent

4. opposite angles are supplementary

(diagonals do not bisect angles or each other)

Midpoint theorem

If connect a line between two lines' midpoints in a triangle, the created line will be parallel to the triangle's other/third line and is half the length of the third side

The sum of the first n positive integers (76!)

Sn=n(n+1)/2

Sum of first n positive/natural even numbers (53!)

n(n+1)

sum of first n positive odd numbers (85!)

n^2

number of variables vs number of equations

if you have 3 variables but only 2 equations, the problem is impossible (think nickel, dime, and quarter problem)