Proving 1 ^ 3 + 2 ^ 3 + 3 ^ 3 + by mathematical induction n^3=（1+2+3+.+n)^2

Proving 1 ^ 3 + 2 ^ 3 + 3 ^ 3 + by mathematical induction n^3=（1+2+3+.+n)^2

When n = 1
Left 1 ^ 3 = 1 right 1 ^ 2 = 1
Left = right
Suppose that the equation holds when n = K
1^3+2^3+3^3+… k^3=（1+2+3+.+k)^2
Then when n = K + 1
1^3+2^3+3^3+… k^3+(k+1)^3
=(1 + 2 + 3 +. + k) ^ 2 + (K + 1) ^ 3 1 + 2 + 3. + k = K (K + 1) / 2 arithmetic sequence
=k^2(1+k)^2/4+(k+1)^3
=(1+k)^2(k^2/4+k+1)
=(1+k)^2(k^2+4k+4)/4
=(k+1)^2(k+2)^2/4
=[(k+1)(k+1+1)/2]^2
=(1 + 2 + 3. + K + K + 1) ^ 21 + 2 + 3 +... K + K + 1 = (K + 1) (K + 1 + 1) / 2 is also an arithmetic sequence
So when n = K + 1, the equation holds
therefore
1^3+2^3+3^3+………… n^3=（1+2+3+.+n)^2

On a map with a scale of 1:6000000, the distance between a and B is 7 cm
The passenger car and the freight car are running from a and B at the same time. The passenger car runs 55 kilometers per hour. After 4 hours, the two cars meet. What is the speed of the freight car

Hello
A. B actual distance between the two places:
7÷1/6000000
=42 million (CM)
=420 (km)
Sum of two vehicle speeds:
420 △ 4 = 105 (km)
Speed of goods vehicle:
105-55 = 50 (km)

If x and y are positive real numbers and X + Y > 2, it is proved that at least one of (1 + x) / Y and (1 + y) / X is less than 2
I hope you can give me some advice,

To the contrary
If they are all greater than or equal to 2, that is, (1 + x) / Y > = 2, (1 + y) / x > = 2,
Then 1 + x > = 2Y, 1 + Y > = 2x. The sum of the two formulas is: 2 + X + Y > = 2x + 2Y, that is, 2 > = x + y, which is contradictory to x + Y > 2, so the assumption is not true, so at least one of (1 + x) / Y and (1 + y) / X is less than 2

A pile of coal weighs 18 tons, and the canteen uses one fourth of the total in three days. How many parts of the total are used in one day, and how many tons are used in three days

One day use = 1 / 4 ／ 3 = 1 / 12 & nbsp; three days eat = 18 × 1 / 4 = 9 / 2 (ton) & nbsp; ~ one moment always 523 answer for you, I wish you progress in your study ~ ~ ~ if you agree with my answer, please click the [adopt as satisfactory answer] button in time ~ ~ ~ the mobile phone questioner comments "satisfied" in the upper right corner of the client, that is

Exercises of arithmetic sequence in mathematics compulsory 5 of senior two
Given the function f (x) = x / 3x + 1, the sequence (an) satisfies A1 = 1, an + 1 = f (an) (n belongs to N +)
1. Find the general term formula of sequence (an)
2. Note Sn = A1A2 + a2a3 + +Anan + 1, find SN

(1) A (n + 1) = f (an) = an / (3an + 1) 1 / a (n + 1) = 1 / an + 3,1 / A1 = 1. Obviously, the sequence {1 / an} is an arithmetic sequence with the first term of 1 and the tolerance of 3. Therefore, 1 / an = 1 / A1 + (n-1) d = 3n-2, so an = 1 / (3n-2). In conclusion, the general formula of sequence {an} is an = 1 / (3n-2). (2

The speed ratio of car a and car B is 5:8. The two cars start from a and B at the same time and meet at 24 kilometers away from the midpoint. How many kilometers are there between the two places? [hint: when a and B meet, the distance ratio and speed ratio are the same]. When the driving time is the same, the distance ratio remains unchanged, and the speed ratio is the same

A: the distance between the two places is 208 km

Given inequality (a-b) x + (3a-b) < 0, the solution is x > - 2
Solving inequality (a-2b) x + (b-2a) ≤ 0

The solution of (a-b) x + (3a-b) ＜ 0 is x ＞ - 2
Knowledge a-b

The two passenger and freight cars leave each other at the same time. When they meet, the distance ratio of the two cars is 6:5. After meeting, the freight car is 12 kilometers faster than the passenger car per hour, and the passenger car is still moving at the original speed. As a result, the two cars arrive at each other's departure station at the same time. It is known that the freight car has been running for 10 hours. How many kilometers is the distance between the two places?

6 △ 5 = 1.2, 12 △ 1.2 × 1.2-1 × 1.2, = 12 △ 3625-1 × 1.2, = 12 △ 1125 × 1.2, ≈ 32.7 (km); distance between the two places: 32.7 × 10 = 327 (km); answer: the distance between the two places is 327 km

Given that 2xa + 3YA + 1 = 0, 2xb + 3yb + 1 = 0, then the expression of the first-order function of the image passing through (XA, ya), (XB, Yb) two points is -?
A and B are used to distinguish

The so-called first-order function of (XA, ya) and (XB, Yb) is to solve an equation so that its two solutions are x = Xa, y = ya, x = XB, y = Yb. Because 2xa + 3YA + 1 = 0 and 2xb + 3yb + 1 = 0, then the two solutions of equation 2x + 3Y + 1 = 0 are exactly Xa, ya, XB, Yb, then 2x + 3Y + 1 = 0

1. Fill in the number according to the rule: 1.4:1.4:9:4.6:16.7:8.25:10 {} {} 2. The number ratio of the two workshops is 4:3
The workshop transferred 54 people to workshop B. the number of people in workshop a and workshop B despised 3:4. How many people were there in workshop a and workshop B?

13 out of 10 and 36, 16 out of 49
Two questions a 216 B 162