Let (3x-1) ^ 4 = ax ^ 4 + BX ^ 3 + CX ^ 2 + DX + e (1) find a + B + C + D + e (2) find a + C + E

Let (3x-1) ^ 4 = ax ^ 4 + BX ^ 3 + CX ^ 2 + DX + e (1) find a + B + C + D + e (2) find a + C + E

Let x = 1
Then a + B + C + D + e = (1-2) ^ 4 = 1
x=0
Then 0 + 0 + 0 + 0 + e = (0-2) ^ 4
e=16
x=-1
Then A-B + C-D + e = (- 1-2) ^ 4 = 81
a+b+c+d+e=1
Add
2(a+c)+2e=82
a+c+e=41

The average of the five natural numbers is 28. None of the five natural numbers is less than 19, and they are not equal to each other. What is the largest number among the five natural numbers?

If you want the biggest number to be as big as possible, you have to make the others as small as possible
So the small four numbers increase from 19: 19 20 21 22, then the remaining one is the largest
28×5 - 20 - 21 -22 -23 = 58 ..
The biggest number is 58

Find the rule: 4, 8, 18~

Find the rule: 4, 8, 18~
4,8,18,22,40,44,70,.(n+1)^2+(-1)^2(n-1)

N ^ 3 + 5N can be divided by which natural numbers, mathematical induction calculation

To solve this problem, we first try to use conjecture (the method of reasonable reasoning) behind our backs,
When n = 1, the original formula is 6, when n = 2, the original formula is 18, when n = 3, the original formula is 42, and when n = 4, the original formula is 84. These numbers can be divided by 6 (don't worry about other natural numbers first), so we guess that n ^ 3 + 5N can be divided by 6
Conjecture: (n ^ 3 + 5N) / 6 is a natural number;
It is proved that: 1) when n = 1, left = 6 / 6 = 1 ∈ n, it is true;
② Suppose n = K (K ∈ n *) (i.e. (k ^ 3 + 5K) / 6 ∈ n)
Then when n = K + 1, the left side = [(K + 1) ^ 3 + 5 (K + 1)] / 6
=(k^3+3k^2+3k+1+5k+5)/6
=(k^3+5k +3k^2+3k +6)/6
=(k^3+5k)/6 +(3k^2+3k)/6 +6/6
=(k^3+5k)/6 +1 + 3k(k+1)/6
=(k^3+5k)/6 +1 + k(k+1)/2
And ∵ K, K + 1 are adjacent natural numbers, so K (K + 1) / 2 ∈ n *, so the left ∈ n *, holds
In conclusion, the conjecture holds
Divisible by 6 means divisible by 1,2,3
So the answer is: 1, 2, 3, 6

It is known that the function f (x) is an even function defined on R, and when x ≤ 0, f (x) = x ^ 2 + 2x, the analytic expression and range of F (x) are written out

When x > 0, - x0, f (x) = x & sup2; - 2x
When x ≤ 0, f (x) = x & sup2; + 2x
When f (x) = x & sup2; - 2x = (x-1) & sup2; - 100, f (x) ≥ f (1) = - 1
f(x)=x²+2x=(x+1)²-1 x

Please look at these numbers:1 ,3,5,______ ,11.What is the missing number in English?

Mathematically, this is an arithmetic sequence with a tolerance of 2. So 5 + 2 = 7; 7 + 2 = 9, 9 + 2 = 11
Therefore, 7 and 9 genera are missing
7 is seven
Nine is nine
So the answer is seven, nine

It is known that a solution of x ^ 2 + kx-2 = 0 is the same as that of the equation 4x-5 / 2x-3 = 0. Find the value of K and find another solution of the equation x ^ 2 + kx-2 = 0

4x-5/2x-3=0
(3/2)x = 3
x = 2
Take x = 2 into equation 1
（2）² + 2k -2=0
2k = -2
k = -1
x²-x-2=0
(x-2)(x+1)=0
x1=2 ； x2=-1
So the other solution is - 1
So, the other solution is - 2

3X-2Y=3M+1
4X-3Y=M+1
The values of X and y of are not positive numbers, and the value range of M is calculated
In particular, the detailed solution of the inequality system, as well as the process of obtaining the value range, if good, add scores!

The key to this problem is simplification
3x-2y=3m+1-----（1）
4x-3y=m+1------（2）
First, we remove X by multiplying (1) by 4 and (2) by 3
12x-8y=12m+4----(3)
12x-9y=3m+3------(4)
(3) Formula minus (4)
Y = 9m + 1 --- (5), because y is not a positive number, so y

The average of new data a, 5, B, 4, - 2 obtained by subtracting 50 from a group of data is 2, and the variance is 8.4

From the mean of a, 5, B, 4, - 2 is 2, and the variance is 8.4, we can get the equation system a + 5 + B + 4-2 = 2 * 5 (1) (A-2) ^ 2 + 9 + (b-2) ^ 2 + 4 + 16 = 8.4 * 5 (2) from formula (1), we can get a = 3-B, bring in formula (2), simplify formula B ^ 2-3b-4 = 0, so B = 4, or B = - 1, when B = 4, a = - 1; when B = - 1, a = 4, so the new data is 4,5, - 1,4

How to subtract 89 times 99 plus 89
2.63 plus 5.8 plus 7.37 plus 4.2 is also a subtraction

89x99+89
=89x(99+1)
=89x100
=8900