Given that an is an equal ratio sequence, A2 = 2, A3 = 1 / 4, then A1A2 + a2a3 +. + Anan + 1= Given that an is an equal ratio sequence, A2 = 2, A5 = 1 / 4, then A1A2 + a2a3 +... + Anan + 1=

Given that an is an equal ratio sequence, A2 = 2, A3 = 1 / 4, then A1A2 + a2a3 +. + Anan + 1= Given that an is an equal ratio sequence, A2 = 2, A5 = 1 / 4, then A1A2 + a2a3 +... + Anan + 1=

[1-(1/8)^n]/28
1-(1/2)^n=(2^n-1)/2^n
The physics problem of electric heating: seeking the electric energy consumed by induction cooker
The thermal efficiency of induction cooker is 90%, and the rated power is 1000W. Xiaoming family needs 8 * 10 ^ 6J to boil water and cook every day
The heat is provided by induction cooker. How much electricity is consumed?
The correct answer is: W electricity = q absorption / thermal efficiency = 8.9 * 10 ^ 6, j = 2.5kW · H
Can heat energy be converted into electricity? W electricity = q absorptive / thermal efficiency? Is this formula right? Tell me what it means?
Yes, all the electric energy is used for heating. Part of it is used for boiling water and cooking, and part of it is lost in the process of utilization~
How is it equal to 2.5?
The square side length is 3. If the side length increases x, the area increases y. The functional relationship between Y and X is______ .
From the square side length 3, the side length increases x, and the increased side length is (x + 3), then the area increases y = (x + 3) 2-32 = x2 + 6x + 9-9 = x2 + 6x
On the positive half axis of x-axis, OA1 = A1A2 = a2a3 = a3a4 = a4a5 is successively intercepted, passing through A1, A2, A3, A4, a5
OA1 = A1A2 = a2a3 = a3a4 = a4a5 is successively intercepted on the positive half axis of x-axis. Perpendicular lines passing through A1, A2, A3, A4 and A5 of x-axis intersect the image of inverse scale function y = 2 / X (x is not equal to 0) at P1, P2, P3, P4 and P5 to obtain right triangle op1a1, a1p2a2, a2p3a3, a3p4a4 and a4p5a5, and their areas are S1, S2, S3, S4 and S5 respectively, then the value of S5 is
Let OA1 = A1A2 = a2a3 = a3a4 = a4a5 = a
Then the coordinate of A5 is (5a, 0), and the abscissa of P5 is 5A
On the image of y = 2 / x, P5 satisfies the equation
So the ordinate of P5 is p5a5 = 2 / (5a)
So S5 = 1 / 2 * p5a5 * a4a5 = 1 / 5
Let OA1 = A1A2 = a2a3 = a3a4 = a4a5 = a
Then the coordinate of A5 is (5a, 0), and the abscissa of P5 is 5A
On the image of y = 2 / x, P5 satisfies the equation
So the ordinate of P5 is p5a5 = 2 / (5a)
So S5 = 1 / 2 * p5a5 * a4a5 = 1 / 5
Calculation of thermal power of physical motor
There is a motor, the rated voltage is 220 V, the normal working current is 25 A, the internal coil resistance is 0.4 ohm
p=I*I*R=250w
Given that the perimeter of a rectangle is 24, (1) then the functional relationship between the area s of the rectangle and the side length a is (2) when a is what, s is the largest?
(1) S = (12-A) a, that is, s = 12a-a ^ 2. (2) formula s = (a-6) ^ 2 + 36, because - (a-6) ^ 2 is less than or equal to 0, so s is less than or equal to 36, when a = 6, s = 36
When a is 6, s is the largest, Smax = 36
S equals the square of a minus 12a
s=axb
2x(a+b)=24
a+b=12
b=12-a
s=ax(12-a)
s=12a-a²
When a = 12 / 2 = 6, s is the largest
s=6x6=36
Prove 2 & sup2; + 4 & sup2; + 6 & sup2; +... + (2n) & sup2; = (2 / 3) n (n + 1) (2n + 1) by mathematical induction
prove:
(1) Let n = 1, (2 × 1) & sup2; = (2 / 3) × 1 × (1 + 1) × (2 × 1 + 1) hold;
(2) Suppose n = K (1 ≤ K ∈ z), the equation holds,
That is 2 & sup2; + 4 & sup2; + 6 & sup2; + +(2k)²=(2/3)k(k+1)(2k+1)；
Then when n = K + 1,
2²+4²+6²+… +(2k)²+[2(k+1)]²
=(2/3)k(k+1)(2k+1)+[2(k+1)]²
=(2/3)(k+1)[k(2k+1)+6(k+1)]
=(2/3)(k+1)[2k²+7k+6)]
=(2/3)(k+1)(k+2)(2k+3)
=(2/3)(k+1)(k+2)[2(k+1)+1]
The equation is also true!
Electric energy consumption of motor
The power of the motor is 38KW, the rated voltage is 380V, and the coil resistance is 0.4 Ω. In one lifting, the motor works normally for 20s, and the heavy object is lifted for 20m. Calculation: 1. The heat generated by the coil during the operation of the motor. 2. In the lifting equipment, if the mechanical efficiency of this lifting is 80%, what is the mass of the lifted object?
P=UI I=P/U=100A
1. The heat generated by the coil during the operation of the motor
Q=I^2Rt=80000j
2.W=UIt-Q=680000J
GH = FW = FH
η=GH/W=0.8 G=27200Kg=27.2t
It is known that the perimeter of a rectangle is 24 cm. (1) write the functional relationship between the area s of the rectangle and the length a of one side. (2) when a is long, s is the largest?
1. The length of one side is a, so the length of the other side is 12-a. s = a (12-A) 2. S ≤ [(a + 12-A) / 2] ^ 2 = 36. If and only if a = 12-A, i.e. a = 6, take the equal sign, so when a = 6, s is the largest
(1) The length of one side is a, and the other side is (12-A) (because the sum of the lengths of both sides * 2 = perimeter), so s = a (12-A) = - A ^ 2-12a (2). The formula of the above function is s = - (a-6) ^ 2 + 36, so when a = 6cm, the maximum s is 36cm2
(1) S = a * (12-A) = 12a-a ^ 2 (2) s = a * (12-A) = 12a-a ^ 2 = - (a-6) ^ 2 + 36 when a = 6, the maximum s is 36
S = 36 When s = a (12-A) a = 6
S is the largest when s = a (12-A) a = 6
S = (12-A) a = the square of 12a - A. if this rectangle with a circumference of 24cm is the largest, then this rectangle must be a square, so a = 24 / 4 = 6cm
When a =? - 2 s / a =? - 6, a =? - 6 S / A is the largest
S = a (12-A) when a = 6, s = 36
S = 12a-a ^ 2 s = (A-3) ^ 2 + 9, so when a = 3, s is the largest.
Let B be the other side, then 24 = 2A + 2B, B = 12-A, s = a * b = a * (12-A) = - A ^ 2 + 12a = - (a-6) ^ 2 + 36, so when a = 6, s max = 36
Satisfy B (n + 1) = BN ^ 2-nbn + 1 and B1 = 2, guess its general formula, and prove it by mathematical induction
Let B (n + 1) = BN ^ 2-nbn + 1 and B1 = 2
Conjecture BN = n + 1
When n = 1
b1=1+1=2
Suppose n = k
bk=k+1
So when n = K + 1
bk+1=（k+1）^2-k(k+1)+1=k+2
Combined with B1 = 2, the hypothesis is proved
If you want to guess, just put in a few numbers
b1=2，
b（n+1）=bn^2-nbn+1
b2=b1^2-b1+1=4-2+1=3
b3=b2^2-2b2+1=9-6+1=4
b4=b3^2-3b3+1=16-12+1=5
b5=b4^2-4b4+1=25-20+1=6
Conjecture BN = n + 1
prove. When n = 1, B1 = 1 + 1 = 2 is satisfied
Let n = k, BK = K + 1
Then n = K + 1... Expansion
b1=2，
b（n+1）=bn^2-nbn+1
b2=b1^2-b1+1=4-2+1=3
b3=b2^2-2b2+1=9-6+1=4
b4=b3^2-3b3+1=16-12+1=5
b5=b4^2-4b4+1=25-20+1=6
Conjecture BN = n + 1
prove. When n = 1, B1 = 1 + 1 = 2 is satisfied
Let n = k, BK = K + 1
Then n = K + 1, BK + 1 = BK ^ 2-kbk + 1 = (K + 1) & # 178; - K (K + 1) + 1 = K & # 178; + 2K + 1-k & # 178; - K + 1 = (K + 1) + 1 also holds
To sum up, BN = n + 1