RULES
Arithmetic sequence:
Tn= a+(n-1) d
general term
𝒏
Sn= [2a + (n-1) d]
𝟐
𝒏
Sn= [a+l]
𝟐
0>a+(n-1) d
when order first negative term.
Geometric sequence:
Tn= arn-1
Sn=
S∞=
general term
𝒂(𝒓𝒏 −𝟏)
𝒓−𝟏
𝒂
𝟏−𝒓
Matrices & determinists:
𝟐
(
−𝟏
𝟑
)
𝟓
Row
Column
Order= Row × Column 2×2
‫ومتتغباش ر‬
‫وتضب هم ف بعض هما كدا شكلهم حلو‬
1
𝟏
A= (
𝟐
−𝟑
)
𝟕
𝟏 𝟐
- At= (
)
−𝟑 𝟕
𝟐 −𝟔
- 2A= (
)
𝟒 𝟏𝟒
𝟏
- A2 = A × A = (
𝟐
- (
[(𝟏 × 𝟏) + (−𝟑 × 𝟐)]
[(𝟐 × 𝟏) + (𝟕 × 𝟐)]
𝟐
𝟓
𝟏
|
𝟔
𝟐
|𝟏
𝟓
𝟏
𝟐
𝟑
|
−𝟑
𝟏
)×(
𝟕
𝟐
−𝟑
) = Row × Column
𝟕
[(𝟏 × −𝟑) + (−𝟑 × 𝟕)]
)
[(𝟐 × −𝟑) + (𝟕 × 𝟕)]
= [(2×6) -(1×5)] = 12-5 = 7
𝟕
𝟐
𝟒| = 2|
𝟑
𝟎
𝟒
𝟏
| -1 |
𝟎
𝟓
𝟒
𝟏
| +7 |
𝟎
𝟓
𝟐
|=
𝟑
Using Cramer’s rule to solve it:
- X+2Y=4
𝟏
Δ=|
𝟐
𝟒
Δx= |
𝟓
𝟏
Δy = |
𝟐
X=
Y=
𝜟𝒙
𝜟
𝜟𝒚
𝜟
=
=
&
2X+Y=5
𝟐
| = (1×1) -(2×3) =-3
𝟏
𝟐
| = (4×1) -(2×5) = -6
𝟏
𝟒
| = (1×5) -(4×2) = -3
𝟓
−𝟔
−𝟑
−𝟑
−𝟑
=2
=1
2
If A= (
𝟓
𝟓
𝟓
Δ=|
𝟓
𝟒
) find A-1 (multiplicative inverse of A)
𝟏𝟎
𝟒
| = 30
𝟏𝟎
𝟏
𝟏 𝟏𝟎
A-1 (
𝜟 −𝟓
𝟏 𝟏𝟎
−𝟒
)= (
𝟑𝟎 −𝟓
𝟓
−𝟐
−𝟒
𝟑
) = (−𝟏
𝟓
𝟏𝟓
𝟏)
𝟔
𝟔
Diffraction:
Y = F(x) ----->
𝒅𝒚
𝒅𝒙
1- If Y = axn -----→
2- If Y= a
3- If Y=
𝟐𝒙𝟓
5-
6- Y=
𝑭𝟏
𝑭𝟐
𝒅𝒚
=0
𝒅𝒙
----→ Y= 3 (2x-5) ---→
4- F1×F2 ----→
√ 𝒙𝒎
= nxaxn-1
𝒅𝒙
a≠0 ----→
𝟑
𝒏
𝒅𝒚
𝒅𝒚
𝒅𝒚
(F1) × F2 + F1 ×
𝒅𝒙 𝒅𝒙
𝒅𝒚 𝒎
-----→
----→
=
𝒅𝒙
𝒅𝒚
𝒅𝒙
−𝟏𝟓 -6
x
𝟐
𝒅𝒚
=
𝒎
−𝟏
𝒏
𝒅𝒙
(F2)
𝑿
𝒏
𝒅𝒚
𝒅𝒚
(𝒇𝟏 )×𝒇𝟐 −𝑭𝟏 × (𝒇𝟐 )
= 𝒅𝒙
7- Y= [F(x)] n ------→
𝒅𝒚
𝒅𝒙
𝒅𝒙
(𝑭𝟐 )𝟐
= n[F(x)] n-1 ×
𝒅𝒚
𝒅𝒙
F(x)
3
‫‪Integration:‬‬
‫‪+c‬‬
‫𝟑‬
‫‪+c‬‬
‫‪+c‬‬
‫𝟑‪𝟑𝒙−‬‬
‫𝟑‪−‬‬
‫𝟐𝒙‬
‫𝟑‬
‫𝟐‬
‫𝟏‪𝒏+‬‬
‫= ‪1- ∫ 𝒂𝒙 dx‬‬
‫𝟏‬
‫𝟐‬
‫= ‪2- ∫ √𝒙 dx = ∫ 𝒙 dx‬‬
‫𝟑‬
‫= ‪3- ∫ 𝟒 dx = ∫ 𝟑𝒙−𝟒 dx‬‬
‫𝒙‬
‫‪∫ 𝟑 dx = 3x+c‬‬
‫‪4- ∫ 𝒂 dx = ax+c‬‬
‫‪+c‬‬
‫‪+c‬‬
‫𝟏‪𝒂𝒙𝒏+‬‬
‫𝒏‬
‫𝟏‪(𝒂𝒙+𝒃)𝒏+‬‬
‫𝒂×)𝟏‪(𝒏+‬‬
‫𝟏‪[𝑭(𝒙)]𝒏+‬‬
‫𝟏‪𝒏+‬‬
‫= ‪5- ∫(𝒂𝒙 + 𝒃)𝒏 dx‬‬
‫𝒚𝒅‬
‫= ‪6- ∫[𝑭(𝒙)]𝒏 F(x) dx‬‬
‫𝒙𝒅‬
‫‪Ex:‬‬
‫𝒙𝒅 𝟓)𝟐 ‪∫ 𝒙(𝟑𝒙𝟐 −‬‬
‫𝟏‬
‫𝒙𝒅 𝟓)𝟐 ‪= ∫ 𝟔𝒙(𝟑𝒙𝟐 −‬‬
‫𝟔‬
‫‪+c‬‬
‫𝟏‬
‫𝟔]𝟐‪[𝟑𝒙𝟐 −‬‬
‫𝟔‬
‫𝟔‬
‫=‬
‫الشخص الذي ميكنه أن يكون أي شخص ويصنع أي شيء سوف يتعرض للنقد والذم وإساءة الفهم‪،‬‬
‫هذا جزء من مثن العظمة‪.‬‬
‫‪4‬‬