Mathematics Dictionary

Mathematics Dictionary
Dr. K. G. Shih

Algebraic Examples

Symbol Defintion

........... Example : LT = less than, x^2 = square of x
Mathematics Dictionary ..... Ennter AL, AN, CA, GC, GE, PM, TR

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P |
Q | R | S | T | U | V | W | X | Y | Z |

Topics by Keywords

Q01. A

Arithematic progression (AP)

AL 27 03 : a^2,b^2,c^2 in AP then 1/(a+b), 1/(a+c),1/(b+c) in AP
AL 27 04 : Find A*a*b*c*(b-c) + B*a*b*c*(c-a) + C*a*b*c*(a-b) if t(i) in AP
- Sum[t(i)] = A for i=1 to a
- Sum[t(i)] = B for i=1 to b
- Sum[t(i)] = C for i=1 to c
AL 08 14 : Angle of triangle ABC in AP, find angles

Arrangement

AL 03 13 : n = n1 + n2 + n3, find arrangement of n1, n2, n3

Asymptote

AL 16 05 : Study the graph of y = ((x - 1)^2)/(2*x)
AL 16 06 : Study the graph of y = x + 2/x^2
AL 27 07 : Study the graph of y = (x^2 + x - 12)/(x^2 - 4)

Go to Begin

Q02. B

Binomial theory

AL 03 14 : 5^(2*n) - 24*n - 1 is divisible by 576

Go to Begin

Q03. C

Go to Begin

Q04. D

Go to Begin

Q05. E

21 11 Equation : Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + x + 1 = 0

AL 11 15 Solution
AL 21 11 Diagram

Equations

AL 11 17 : x^4 + b*x^3 + c*x^2 + d*x + e = 0 has roots 1,2,3,4
AL 11 18 : x^4 - 10*x^3 + 35*x^2 - 50*x + 24 = 0 has roots 1,2. Find others

06 11 Exponent : e^x + e^(2*x) + e^y + e^(2*y) = 12

06 11 Sketch
06 12 Find y if x = ln(2)

Go to Begin

Q06. F

01 04 Factor theory
07 01 Factorial n!
07 09 Fibonacci's sequence in Pascal triangle
05 04 Focus of parabola

Go to Begin

Q07. G

AL 08 02 : Geometric series
AL 08 15 : Angles of triangle ABC in GP, find angles

Go to Begin

Q08. H

06 17 Hyperbolic functions : Identities

Prove that cosh(x)^2 - sinh(x)^2 = 1
Prove that sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)
Prove that sinh(2*x) = 2*sinh(x)*cosh(x)

17 09 Hypergeometric probability H(x) = C(n,x)*C(N-n,n-x)/C(N,n)

17 10 Hypergeometric probability H(x) = C(n,x)*C(N-n,s-x)/C(N,s)
Go to Begin

Q09. I

Induction method

14 00 Induction method
14 06 Induction : S(n) = 1/(1*3) + 1/(3*5) + .... = (1 - 1/(2*n+1))/2
14 07 Induction : S(n) = 1/(1*2*3) + 1/(2*3*4) + .... = n*(n+3)/(4*(n+1)*(n+2)
14 08 Induction : S(n) = 1^2 + 3^2 + 5^2 + ... = n*(4*n^2 - 1)/3
14 09 Induction : S(n) = 1*(2^2) + 2*(3^2) + ... = n*(n+1)*(n+2)*(3*n+5)/12
14 10 Induction : S(n) = (1 -4/1)*(1 - 4/9)*(1 - 4/25)*(1 - 4/49)*......
14 11 Induction : S(n) = (1 + 1)*(1 + 1/2)*(1 + 1/3)*.....*(1 + 1/n) = n + 1
14 12 Induction : S(n) = (1^2)/(1*3) + (2^2)/(3*5) + .... = (n*(n+1))/(2*(2*n+1))
14 13 Induction : S(n) = 2 + 4 + 6 + ..... + 2*n = (n + 1/2)^2. Is it true
14 14 Induction : S(n) = n^2 + n + 41. Is it always a prime number ?
14 17 Induction : Prove that 1! + 2! + 3! + 4! + ..... + n! = 3^(n-1)

15 00 Inequality

15 02 Solve x^2 - 6*x + 8 < 0
15 08 Solve (x-1)/(x+1) > 1
15 09 Solve (x-1)/(x+1) > 2

Inverse

06 08 Inverse of y = e^x
09 12 Inverse of hyperbolic functions
04 03 Inverse of linear functions
09 03 Inverse of y = ln(x)
05 03 Inverse of quadratic functions

Go to Begin

Q10. J

Go to Begin

Q11. K

Go to Begin

Q12. L

Linear equations

AP 04 11 : Sketch y = Abs(x - 1) + Abs(x + 1)
AL 04 12 : Solve Abs(x - 1) + Abs(x + 1) = 3

09 02 Logarithmic laws
Go to Begin

Q13. M

AL 03 04 Magic number patterns

AL 03 05 Matrix pattern : Find row number and column number of 100

1, 03, 06, 10, 15, 21, ...
2, 05, 09, 14, 20, .......
4, 08, 13, 19, ...........
7, 12, 18, ...............

AL 03 10 : Multiple of 49 whose digits are same.
Go to Begin

Q14. N

17 12 Normal distribution : N(z LT a)
Numbers
- 03 11 : Amicable number pairs 220 and n, find n.
- 03 05 : Numbers arranged in matrix. 1st row sequence 1,3,6,10, ...
- 03 12 : Perfect numbers - Find 3rd perfect number
- 03 03 : Properties based on factors of number.
- 14 15 : Prove that ((n^3) + 3*(n^2) + n)/3 is an integer
- 14 16 : Prove that (2*(n^3) + 3*(n^2) + n) is divisible by 6
07 01 n! : definition and trailor zeros in n!

Go to Begin

Q15. O

Go to Begin

Q16. P

05 04 Parabola : Locus, Focus and directrix
08 05 Pascal triangle : Sequence and series
03 12 Perfect number : How to find the third perfect number ?
03 07 Perfect square numbers.
17 01 Permutation : P(n,r)
- 17 02 Take r from N symbols for arrangement without duplicate : P(n,r)
- 17 02 Take r from N symbols for arrangement with duplicate : n^r
08 13 Pi in series
21 00 Polynomial equation
- 21 11 Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + x + 1 = 0

Go to Begin

Q17. Q

11 04 Quaint equation
21 01 Quaint equation : Graphic solution
12 06 Quartic equation : (x-p)*(x-q)*(x-r)*(x-s) = t
11 03 Quartic equation : (x-5)*(x-7)*(x+4)*(x+6) = 504
21 01 Quartic function : y = a*x^4 + b*x^3 + ....
21 01 Quartic function : y = (x-p)*(x-q)*(x-r)*(x-s) - t
11 01 Quadratic equations
11 10 Quadratic equations : x^2 -5*x + 2*Sqr(x^2 - 5*x + 3) = 12
05 01 Quadratic functions : Defintion and application
05 02 Quadratic functions : With absolute operation
05 03 Quadratic functions : Inverse
05 04 Quadratic functions : Parabola

Go to Begin

Q18. R

Rational function

AL 16 01 : y = 1/(x + 1)
AL 16 02 : y = 1/(x^2 - 1)
AL 16 03 : y = 1/(x^3 -2*x - x + 2)
AL 16 06 : y = x + 4/(x^2)
AL 16 04 : y = (x^2 -2*x + 1)/(x)
AL 16 05 : y = ((x - 1)^3)/(2*x)
AL 27 05 : y = (x^2 + x - 12)/(x^2 -4)

03 01 Real number system

01 04 Remainder theory
Go to Begin

Q19. S

27 01 Sequence : 1,3,7,13,21,....
07 04 Sequence in Pascal triangle.
- T(1)
- T(n)
- T(n*(n+1)/2!)
- T(n*(n+1)*(n+2)/3!)
- T(n*(n+1)*(n+2)*(n+3)/4!)
08 01 Series : A.P.
08 02 Series : G.P.
08 13 Series : pi
Series
- AL 08 03 Series : Sum[n^2] = n*(n+1)*(2*n+1)/6.
- AL 08 04 Series : Sum[n^3] = (n*(n+1)/2)^2.
- AL 07 08 Series : Sum[n^4] = ?
Series : Series and sequence in Pascal triangle.
- AL 08 05 : Sum[n*(n+1)/2!] = n*(n+1)*(n+2)/3!
- AL 08 06 : Sum[n*(n+1)*(n+2)/3!] = n*(n+1)*(n+2)*(n+3)/4!
- AL 08 07 : Sum[n*(n+1)*(n+2)*(n+3)/4!] = n*(n+1)*(n+2)*(n+3)*(n+4)/5!
Series and sequence in Pascal triangle.
- AL 08 05 : Sum[C(n+1),2)] = C(n+2,3)
- AL 08 06 : Sum[C(n+2),3)] = C(n+3,4)
- AL 08 07 : Sum[C(n+3),4)] = C(n+4,5)
AL 08 06 Series : Special
- S(n) = 1/(1*2) + 1/(2*3) + 1/(3*4) + ........ + 1/((n-1)*n) = ?.
- S(n) = 1/(1*2*3) + 1/(2*3*4) + ... + 1/((n*(n+1)*(n+2)) = ?
- S(n) = 1/3 + 1/15 + 1/35 + ...... + 1/((2*n-1)*(2*n+1)) = ?
08 08 Series : 1^3 + 3^3 + 5^3 + ..... (2*n-1)^3 = ?
08 09 Series : 1*(2^2) + 2*(3^2) + 3*(4^2) + ..... n*((n+1)^2) = ?
08 07 Series : S(n) = 1 - 2 + 3 - 4 + 5 - 6 + ....... - n if n is even
08 08 Series : 1^3 + 3^3 + 5^3 + .... = ?
08 09 Series : 1*(2^2) + 2*(3^2) + 3*(4^2) + .... = ?
08 10 Series : 1^3 + 2^3 + ... + n^3 GT (n^4)/4 GT 1^3 + 2^3 + ... +(n-1)^3
08 11 Series : (1-1/4)*(1-1/9)*(1-1/16)*....*(1-1/((n+1)^2 = (n+1)/2*n. n GT 1
08 12 Series : 1 - 2 + 3 - 4 + ...... + n and n is odd
11 10 Simultaneous equations
- x + y = 3 (line)
- x*y = 2 (Hyperbola)
11 10 Simultaneous equations
- x - y = 1
- x*y = 2 (Hyperbola)
11 10 Simultaneous equations
- x^2 + y^2 = 4 (Circle)
- x*y = 1 (Hyperbola)
16 06 Slant asymptote in y = x + 4/(x^2)
03 07 Square free numbers.
03 08 Square root : How to find Sqr(3) ?