Theta

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32 terms · Theta junk you need to know

|a+bi|

√[(a^2)+(b^2)]

Number of digits in x^y

(Ex. 7^43)

y(logx)+1

43(log7)+1
36.339...+1
37.339...
37 (Eliminate the decimal part)

"k"th Triangular Number

T(k)=[k(k+1)]/2

"n"th Pentagonal Number

P(n)=[n(3n-1)]/2

Sum of First "n" Perfect Squares

[n(n+1)(2n+1)]/6

Sum of First "n" Perfect Cubes

[(n/2)(n+1)]^2

What is the determinant of a singular matrix?

0
That is why singular matrices have no inverses.

For any square matrix A,
A • A^(-1)

The Identity Matrix
| 1 0 |
| 0 1 | For a 2x2 matrix,

| 1 0 0 |
| 0 1 0 |
| 0 0 1 | For a 3x3 matrix, etc.

For any square matrix A,
|A| - |A^T|
(A^T is the transverse of the matrix A)

0
The determinant of any square matrix is equal to the determinant of its transverse matrix.

Area of a Triangle Using Matrices

1 | x(1) y(1) 1 |
±— | x(2) y(2) 1 |
2 | x(3) y(3) 1|

1/2 is indicated as ± so that you can multiply whichever number would result in a positive value, depending on the situation, because area must be positive.

x(n) or y(n) means x "sub-n" or y "sub-n"

Arithmetic Mean of a and b

(a+b)/2

For a, b, and c, the arithmetic mean is
(a+b+c)/3 etc.

Geometric Mean of a and b

√(ab)

For a, b, and c, the geometric mean is
∛(abc) etc.

Harmonic Mean of a and b

(2ab)/(a+b)

For a, b, and c, the harmonic mean is
(3abc)/(a+b+c)

Base Changing

Sorry, I don't know how to explain this one on here you just have to know it.

Number of Positive Integral Factors of a Number (Ex. 3240)

Find the prime factorization of the number and multiply the exponents of the prime factors after adding one to each of them.

3240 = 2^3 • 3^4 • 5
(3+1)(4+1)(1+1)
4 • 5 • 2
40

Binomial Theorem

Sorry, again, I don't know how to explain this one on here you just have to know it.

Circular Permutations (keyring)

(n-1)!/2

Circular Permutations (non-keyring)

(n-1)!

√n+√(n+√(n+...=x

Square both sides
Set the radical junk equal to x
Solve the quadratic equation
The positive solution is the solution to the entire equation

Find A+B / Find AB
√(8-2√15)=√A+√B

Square both sides
8-2√15=A+B+2√AB
A+B=8
AB=15

Sum of the Roots of a Quadratic Equation

ax^2 + bx + c = 0
-b/a

Product of the Roots of a Quadratic Equation

ax^2 + bx + c = 0
c/a

Sum of the Reciprocals of the Roots of a Quadratic Equation

ax^2 + bx + c = 0
-b/c

Sum of Roots/Product of Roots
(-b/a)/(c/a)
-b/c

Sum of the Squares of the Roots of a Quadratic Equation

ax^2 + bx + c = 0
(b^2 - 2ac)/a^2

Sum of the Squares of the Reciprocals (or the Reciprocals of the Squares) of the Roots of a Quadratic Equation

ax^2 + bx + c = 0
(b^2 - 2ac)/c^2

Sum of the Roots of Other Polynomial Equations

Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0
-B/A

Product of the Roots of Other Polynomial Equations

Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0
If n is even, K/A
If n is odd, -K/A

Sum of the Reciprocals of the Roots of Other Polynomial Equations

Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0
-J/K

Sum of the Squares of the Roots of Other Polynomial Equations

Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0
(B^2 - 2AC)/A^2

Number of Zeroes at the End of a Specific Number Factorial
Ex. 1000

Divide the number by the largest integral power of 5 less than the original number
Continue dividing by decreasing integral powers of 5 until you reach 5
Add the quotients

Ex.
1000÷625=1
1000÷125=8
1000÷25=40
1000÷5=200
1+8+40+200=249
249 zeroes

"p"th Even Perfect Number

(2^p - 1)•(2^(p-1))
Where p is a prime number

In a rectangle with the dimensions x by y, the number of distinct paths from one corner to the other

(x+y)!/x!y!

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