How to find the first n terms of the square of sequence an = n and Sn = 1 + 4 + 9 + 16 + 25 + (n) 2, where 2 is the square

How to find the first n terms of the square of sequence an = n and Sn = 1 + 4 + 9 + 16 + 25 + (n) 2, where 2 is the square

Formula 1 + 4 + 9 + 16 + 25 + (n) 2 = 1 / 6 * n (n + 1) (2n + 1)

If the factorization result of polynomial x ^ 2-4x + m is (x + 3) (x-n), then M / N is equal to
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x^2-4x-21=(x+3)(x-7)
m/n=-3

A & # 178; + 2B & # 178; = 6. Find the minimum value of a + B,

Solution 1: discriminant method
Let a + B = t, then a = t-b. [1]
The substitution condition is: (T-B) ^ 2 + 2B ^ 2 = 6,
3b^2-2tb+(t^2-6)=0.[2]
∵ B is a real number, Δ ≥ 0,
That is, 4T ^ 2-12 (T ^ 2-6) ≥ 0,
It is reduced to T ^ 2 ≤ 9,
∴-3≤t≤3.
When t = - 3, B = - 1 is obtained from [2], and a = - 2 is obtained by substituting [1]
So the minimum value of a + B is - 3 (when a = - 2, B = - 1)
Solution 2: trigonometric substitution
a^2+2b^2=6→(a^2)/6+(b^2)/3＝1,
Let a = (root 6) cosx, B = (root 3) SiNx, where x ∈ R
A + B = (root 3) SiNx + (root 6) cosx
=Under the root sign [(root 3) ^ 2 + (root 6) ^ 2] sin (x + θ)... [1]
=3sin (x + θ), (where θ is the auxiliary angle)
The minimum value of sin (x + θ) is - 1,
So the minimum value of a + B is - 3
Note: formula [1] uses the formula: asinx + bcosx = root (a ^ 2 + B ^ 2) * sin (x + θ),
The "auxiliary angle θ" satisfies the condition "Tan θ = B / a", and the quadrant position of auxiliary angle θ is determined by the quadrant position of point (a, b)

It is known that the area of a parallelogram is 144, and the heights of two adjacent sides are 8 and 9 respectively, then its perimeter is______ ．

∵ the area of a parallelogram is 144, the heights of the two adjacent sides are 8 and 9, the lengths of the two adjacent sides are 144 △ 8 = 18144 △ 9 = 16, and its circumference is 18 + 16 + 18 + 16 = 68

Solving equation 1 / (X-2) (x-3) - 3 / (x-1) (x-4) + 1 / (x-1) (X-2) = 1 / (x-4)
The required rule: 1 / N (n + 1) = 1 / n-1 / (n + 1)

1 / (X-2) (x-3) - 3 / (x-1) (x-4) + 1 / (x-1) (X-2) = 1 / (x-4) 1 / (x-3) - 1 / (X-2) - 1 / (x-4) + 1 / (x-1) + 1 / (X-2) - 1 / (x-1) = 1 / (x-4) 1 / (x-3) - 1 / (x-4) = 1 / (x-4) 1 / (x-3) = 2 / (x-4) 2 (x-3) = x-42x-6 = x-4x = 2, so the original fractional equation has no solution

The surface areas of p-abc are s △ ABC = 6, s △ PAB = 3, s △ PBC = 4, s △ PCA = 5, and the dihedral angles of each side and bottom are equal
Find the dihedral angle

Let the dihedral angle of the side face and the bottom face be a. according to the area projection theorem, the area of a plane figure projected on a plane is equal to the area of the original figure multiplied by cos (included angle)
The sum of the projected areas of the three sides on the bottom is equal to the bottom area
6 = (3 + 4 + 5) * cosa, cosa = 1 / 2
Therefore, the dihedral angle is 60 degrees

The solution equation is 1.12 / (0.5x-1) = 42.12x-3 = 8x + 17 3.25.2x/3 = 6.3 * 4.9.8 * one and half - x * 50% = 2.4

1.12/(0.5X-1)=4
0.5x-1=12/4
x=8
2.12x-3=8x+17
4X=20
x=5
3.25.2x/3=6.3
x=6.3*3/25.2
x=0.75
4.9.8*(3/2)-x*0.5=2.4
x=(14.7-2.4)/0.5
x=24.6

If the variables X and y satisfy the constraint conditions {3 ≤ 2x + y ≤ 9,6 ≤ X-Y ≤ 9, then the minimum value of Z = x + 2Y is? Why - 6,

6≤x-y≤9
=> -9≤y-x≤-6
Then from: 3 ≤ 2x + y ≤ 9
Two infinitives are added to obtain:
-9+3 ≤（y-x)+(2x+y)≤-6+9
The results show that - 6 ≤ (x + 2Y) ≤ 3
So, the minimum value of Z = x + 2Y is - 6

Given that a and B are reciprocal, C and D are opposite, and M is the largest negative integer, try to find the value of M / 3 + 4C + 4d-23 / 8ab-5

m=-1
ab=1
c+d=0
-1/3+[0-23]/3=-8

It is known that X1 and X2 are the two roots of the equation 2x & # 178; - 3x-1 = 0. Using the relationship between the root and the coefficient, we can find the values of the following formulas
x2²÷x1

We can solve X1 and X2 with the formula of finding roots. There are two cases in this expression, one is big root when x1, the other is small root when x1