algebra and geometry tips
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30 May 2020, 21:14
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1. A positive integer n has an odd number of factors, then n is a perfect square
2. The formula for solving the number of factors: decompose the integer n into the product of prime factors, and then multiply each prime factor by one. N = a * a * a * b * b * c then the number of factors = (3 + 1) (2 + 1) (1 + 1)
eg. 200 = 2 * 2 * 2 * 5 * 5 number of factors = (3 + 1) (2 + 1) = 12
3. The sum of the last three digits of the number divisible by 8 can be divisible by 8; the sum of the digits of the number divisible by 9 can be divisible by 9. The number of divisible by 3, the sum of each bit can be divisible by 3.
4. Polygonal internal angle = (n-2) x180
5. Diamond area = 1/2 x diagonal product
6. Euler's formula: number of edges = number of faces + number of vertices -2
7. Triangle Yuxuan Theorem
C2 = A2 + B2-2ABCOSβ, β is the angle between AB lines
8. Sine theorem: A / SinA = B / SinB = C / SinC = 2R (A, B, C are the sides and corresponding angles, R is the radius of the circumscribed triangle)
9.Y = k1X + B1, Y = k2X + B2, the condition of two lines perpendicular is K1K2 = -1
10.N's factorial formula:
N! = 1 * 2 * 3 * .... (N-2) * (N-1) * N and specify 0! = 1 1! = 1
Eg: 8! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8
11. Get familiar with the values of root numbers 2, 3, and 5
sqrt (2) = 1.414 sqrt (3) = 1.732 sqrt (5) = 2.236
12. ... 2/3 as many A as B: A = 2/3 * B
... twice as many ... A as B: A = 2 * B
13. Conversion between Fahrenheit and Celsius
Conversion formula: (F-32) * 5/9 = C
PS. Conversion of common measurement units: (Check the appendix of the Oxford Dictionary yourself)
Exercises:
1: There are sequence problems: a1 = 2, a2 = 6, an = an-1 / an-2, find a150.
Solution: an = an-1 / an-2, so an-1 = an-2 / an-3, bring the former formula to an = 1 / an-3, and then split it again to get an = an-6, also That is to say, this sequence is based on 6 cycles, then a150 = a144 = ... = a6, using a1, a2 can calculate a6 = 1/3.
If you really can't think of this method, you can write a few items and see soon that a150 = a144, boldly speculate that the sequence is obtained with a period of 6, and then write a1-a13 (that is, you can see the law when you write it), It is not difficult to find that a6 = a12, a7 = a13, and then, if you count a little, you can know that a150 = a6, which is also calculated as 1/3.
2: Ask the degree of increase of 30 degrees Fahrenheit and the ratio of 62.
key: F = 30 * 9/5 = 54 <62
3: The problem of the Fibonacci sequence: Known, a1 = 1 a2 = 1 an = an-1 + an-2, ask the average of a1, a2, a3, a6 and a1, a3, a4 , a5 comparison of the average size of the four items.
Answer: The Fibonacci sequence is the third term is the sum of the first two, and so on to get a1-a6 as:
1 1 2 3 5 8 13 21 a1 + a2 + a3 + a6 = 12, a1 + a3 + a4 + a5 = 11, so it is greater than.
4: How many integer pairs (x, y) satisfy x ^ 2 + y ^ 2 <= 100?
key: Write out in the order of the possible circumstances of X:
X = Y =
1-9
1-9
1-9
1-9
1-8
1-8
1-7
1-6
1-4 => Myanswer: add up = 69
5: 24, 36, 90, 100, the number is divided by all its prime factors. The final result is the prime number: Key: 90
6: 0.123456789101112 ..., this decimal lists all integers indefinitely without looping. What is the 100th digit after the decimal point?
Key: number of digits
1 2 3 4 5 6 7 8 9 10
10 11 12 ………………………… 19 20
20 21 ……………………………… 29 20
30 .................................... 39 20
40 ……………………………… 49 20
50 51 52 53 54 55 56 —————— 101st = 5 ??
7: 2904x = y2 (y squared), x and y are positive integers, find the minimum value of x.
Because: X ^ 2 × Y ^ 2 × Z ^ 2 = (X × Y × Z) ^ 2
So divide 2904 by 2 = 2 × 2 × 2 × 3 × 11 × 11 = 2 ^ 2 × 11 ^ 2 × 6, then multiply by 6 and it ’s OK
2 ^ 2 × 11 ^ 2 × 6 × 6 = (2 × 11 × 6) ^ 2 = 132 ^ 2
Key: The smallest x = 6
8: The sequence An = 1 / n-1 / (n + 1), n> = 1, ask for the sum of the first 100 items.
Answer: An = 1 / n-1 / (n + 1)
An-1 = 1 / (n-1) -1 / n
An-2 = 1 / (n-2)-/ (n-1)
………………………
………………………
A1 = 1-1 / 2
Adding the left side is An + An-1 + …… + A1 = 1-1 / (n + 1) ... the result after eliminating many and many items
Key: Bring n = 100 into the sum of the first 100 items as 100/101
9: Isosceles triangle with waist 6. The height on the bottom edge is x and the bottom edge is y. Ask who is 4x2 + y2 and 144.
Answer: The Pythagorean Theorem gives (y / 2) 2 + x2 = 62, so 4x2 + y2 = 144
10: -1 <R <T p question: the relationship between r + r * t * t and -1 <