Help me find a pattern, thank you!!

The first two digits are added to get the third digit. 1+2=3，2+3=5，3+5=8

a(n+1) = a(n) +a(n-1)

Proof of the Perbonacci sequence.

2006-4-29 12:35:50 |by: source less ].

Pebonacci sequence: 1, 1, 2, 3, 5, 8, 13,。。

The recursive formula for the Perbonacci sequence: f(n+2) = f(n+1) +f(n).

f(1)=f(2)=1。

Its general term is solved as follows:

f(n+2) = f(n+1) +f(n) => f(n+2) -f(n+1) -f(n) = 0

Let f(n+2) -af(n+1) = b(f(n+1) -af(n)).

f(n+2) -a+b)f(n+1) +abf(n) = 0

Obviously a+b=1 ab=-1

From Veda's theorem, we know that a and b are the two roots of the quadratic equation x 2 - x - 1 = 0.

The solution gives a = (1 + 5) 2, b = (1 - 5) 2 or a = (1 - 5) 2, b = (1 + 5) 2

Let g(n) = f(n+1) -af(n), then g(n+1) = bg(n), and g(1) = f(2) -af(1) = 1 - a = b, so g(n) is an equal proportional series, g(n) = b n, i.e.

f(n+1) -af(n) = g(n) = b^n --1)

In equation (1), the above two sets of solutions of a b are substituted respectively, and because of the symmetry, x = (1 + 5) 2 and y = (1 - 5) 2 are set to obtain:

f(n+1) -xf(n) = y^n

f(n+1) -yf(n) = x^n

Subtract the above two formulas to get:

x-y)f(n) = x^n - y^n

f(n) = (x^n - y^n)/(x-y) = /√5

The first two numbers are added to the third number.

The answer is 175.

So, you see, if you subtract the ordinal number from each number, you get 4, 9, 25, 49, 121. In order, it is 2, 3, 5, 7, 11 squared, that is, the square of each continuous prime number, and the next one is 13 squared + 6 = 175

6 and 2 are 8 apart, and -2 and 6 are also 8 apart

Results 10 14 18

I'm tired, I don't know if I can see this ** clearly, and I don't know if I have explained it clearly, hopefully

Derivation process: The analysis shows that the number in the middle of the previous row minus the number on the left side of the next row is equal to the number on the right side of the next row (e.g., 1 3-1 4=1 12, 1 4-1 5=1 20).

Therefore, the backward method is used (which can be reversed to the 5th row from the bottom, i.e. row m-4).

Line m-4: 1 m-4

Line M-3: 1 m-3 1 (m-4)(m-3).

Row m-2: 1 m-2 Bochun 1 (m-3)(m-2) 2 (m-4)(m-3)(m-2).

Line m-1: 1 m-1 Lu Min 1 (m-2) (m-1) 2 (m-3) (m-2) (m-1) 6 (m-4) (m-3) (m-2) (m-1).

Row m: 1 m 1 m(m-1) 2 (m-2)(m-1) 6 m(m-1)(m-2)(m-3) 24 m(m-1)(m-2)(m-3)(m-4).

Up to the nth digit is: (n-1)(n-2)(n-3)....1/m(m-1)（m-2）(m-3)(m-4)…(m-n+1)

Use the permutations and combinations learned in high school until: (n-1) (n-2) (n-3)....1=(n-1)!

m m(m-1)（m-2）(m-3)(m-4)…(m-n+1) = am n (m is below, n is above).

So the nth number in row m is: (n-1)!/am n

You have to do the math on the scratch paper by yourself in this question, mainly using the permutations and combinations knowledge of high school, using the method of backwards, and you can look at the number of high school if you are not familiar with it.

Using the identity (n+1) 3=n3+3n2+3n+1, we get:

n+1) 3-n3=3n2+3n+1,n3-(n-1) 3=3(n-1) 2+3(n-1)+1...

Add the two ends of this n equation separately to get:

n+1) 3-1=3(12+22+32+..n2)+3(1+2+3+..n)+n, due to 1+2+3+.n=(n+1)n 2, the above formula gets:

n 3+3n2+3n=3(12+22+32+..n2)+3(n+1)n/2+n

1 squared + 2 squared + 3 squared +...The square of n = nx(n+1)x(2n+1) of 6

Regularity: 1x1+2x2+...nxn=1/6n(n+1)(2n+1)

It's a mathematically overturned formula.

1²+2²+…n = n (n+1) (n+2) I don't know if it's right or not. But I can tell you that the square can be spelled out directly with Sogou :-d

1 question 103050702040608

102 is divided into two parts, one is 1 and the other is 02, 1 is followed by the odd number 03, and 02 is followed by the even number 04, so that the second number 1030204 is obtained; By analogy, the odd number 05 is added after 03, and the even number 06 is added after 04, so that the third number is obtained; 05 followed by the odd number 07, 06 followed by the even number 08, and I get my answer 103050702040608

2 Question 7 5

Because the first number divided by the second number equals the third number, i.e. 35 7 = 5; The second number divided by the third number equals the fourth number.

1 question, add a 03 before 02, and add 04 after 02:; Then add 05 before 02, and add 06 after 04; Then add 07 before 02 and 08 after 06, so the answer is 103050702040608

Question 2 35 divided by 7 equals 5, 7 divided by 5 equals 7/5 5 divided by 7/5 equals 25/7 So the answer is 7/5

Khan: This is a series of equal differences.

The latter number is greater than the previous number 2 If a number is x, the other numbers are x+ or -2 multiplied by n, add or subtract, this symbol can't be played, subtract below the + sign, you should understand.

n Absolutely.

Let's say the first number is x

The formula is x+(x+2)+(x+4)+(x+6)=172 to calculate, and you know x 40

Then these four numbers are 40 42 44 46

The fourth question can be calculated, or the formula x+(x+2)+(x+4)+(x+6) 2008

Calculate it out, x 499

So I can't.

Because it's an even number.

Got it? Brothers.

One-quarter, three-seventh, minus five-tenths, seven-thirteenths, ninety-sixteenths, minus eleven-nineteenths, (13/22), (15/25), (minus 17/28)...

1 6,1 Qingxiao Xiao2,(3 2),4 Shenglu 1 2,13 1 21 2,3 1 2,7,10,20,(23),(46)3 1,150%,1,3 4,60%, (decimal (21 40) fraction (51%) percent.

90%, 1, 4, 5, 110%, 5) fraction (60%), percentage (decimal, 1 2,140%.

3 20, (40%) percent, , 9 10, 115% (decimal (33 20) fraction (190%) percent.