计そΑ

Ωよ祘

1.In a quadric equation ax²+bx+c=0,

  x=-b∮≡(b²-4ac)

               2a

=≡(b²-4ac)(it is called discriminant)

When is greater than 0, the equation will have two distinct real roots.

When is equal to 0, the equation will have two same real roots.

When is smaller than 0, the equation will have no real roots.(or complex root)

Let the solution of x be x₁ and x₂

then, x₁+x₂=-b/a, x₁x₂=c/a

单畉计

1.Consider a sequence that starts with the first term A₁, and A₂-A₁=A₃-A₂=A₄-A₃=...=A_n-A_n-1. This kind of sequence is called Arithmetic Progression(A.P.).

  Consider another A.P., let its first term and common difference be a and d respectively. Then the first few terms are:

  a, a+d, a+2d, a+3d,...

  To find a formula for T_n, rewrite the A.P.:

  a+(1-1)d, a+(2-1)d, a+(3-1)d, a+(4-1)d,...

So, T_n=a+(n-1)d

2.And the sum(S_n) from a to a+(n-1)d(l) is :

S_n=½n[2a+(n-1)d]

Because the last term l=a+(n-1)d

So, an alternative formula for S_n is :

S_n=½n(a+l)

3.The arithmetic mean(A.M.) is :

  b=½(a+c)

单ゑ计

1.Consider a sequence that starts with the first term A₁, and A₂/A₁=A₃/A₂=A₄/A₃=...=A_n/A_n-1. This kind of sequence is called Geometric Progressions(G.P.).

  Consider a G.P.

  Let its first term and common ratio be a and r respectively. Then the first few terms are:

  a, ar, ar², ar³,...

  To work our a formula for T_n, rewrite the G.P. as below:

  ar^1-1, ar^2-1, ar^3-1, ar^4-1

So, T_n=ar^n-1

2.And the sum (S_n) from a to ar^n-1 is :

S_n=a(1-rⁿ)/(1-r) or S_n=a(rⁿ-1)/(r-1)

3.The Geometric Mean(G.M.) is:

  b=∮≡ac

兜Α

1.For f(x)=0

  if f(a+bi)=0(a,bR & b is not equal to 0)

  then, f(a-bi)=0

2.for f(x)=0

  if f(a+≡b)=0(a,bR & b is not equal to o)

  then, f(a-≡i)=0

3.For f(x)=0

  The remainder of f(x)/(ax-b) is equal to f(b/a)

箉﹚瞶

1.For f(x)=A_nxⁿ+...+A₂x²+A₁x+A₀

  f(x)[x], A_n is not equal to 0

  if f(x)=Q(x)(px-q)

  then, p|A_n, q|A₀

2.Let f(x)=(ax-b)Q(x)

   1. if x=1, f(1)=(a-b)Q(1), then (a-b)|f(1)

   2. if x=-1, f(-1)=-(a+b)Q(-1), then (a+b)|f(-1)